# Thomson backscattering from laser-generated, relativistically moving high-
density electron layers

A Paz1,2, S Kuschel1, C Rödel1,2, M Schnell1,2, O Jäckel1,2, M C Kaluza1,2 and
G G Paulus1,2 1Institut für Optik und Quantenelektronik, Friedrich-Schiller-
Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany  2Helmholtz Institut
Jena, Fröbelstieg 3, 07743 Jena, Germany<EMAIL_ADDRESS>
###### Abstract

We show experimentally that XUV radiation is produced when a laser pulse is
Thomson backscattered from sheets of relativistic electrons which are formed
at the rear-surface of a foil irradiated on its front side by a high-intensity
laser. An all-optical setup is realized using the Jena Titanium:Sapphire TW
laser system (JETI). The main pulse is split into two pulses: one to
accelerate electrons from thin aluminium foil targets to energies of the order
of some MeV and the other, counterpropagating probe pulse is Thomson-
backscattered off these electrons when they exit the target rear side. The
process produced photons within a wide spectral range of some tens of eV as a
result of the broad electron energy distribution. The highest scattering
intensity is observed when the probe pulse arrives at the target rear surface
100 fs after the irradiation of the target front side by the pump pulse,
corresponding to the maximum flux of hot electrons at the interaction region.
These results can provide time-resolved information about the evolution of the
rear-surface electron sheath and hence about the dynamics of the electric
fields responsible for the acceleration of ions from the rear surface of thin,
laser-irradiated foils.

###### pacs:

41.75.Jv, 52.25.Os, 52.38.Kd, 52.27.Ny

††: New Journal of Physics

##  1 Introduction

Thomson scattering of intense laser pulses from relativistic electrons has the
potential to produce high-quality, ultrashort pulses of well-collimated x-ray
radiation [1, 2, 3]. Such an x-ray source would have a great impact on
biomedical imaging, spectroscopy and the analysis of ultrafast structural
dynamics [4, 5]. When implemented using an all-optical setup such that the
laser pulse scatters from laser-accelerated electrons, it possesses the
additional advantage of having a lower electron energy requirement to generate
a given photon energy compared to undulator-based radiation sources at
conventional electron accelerators such as synchrotrons. This is due to the
fact that the laser wavelength is orders of magnitude shorter than the
undulator magnet period [1]. It is also compact in size and provides better
control over the synchronization as compared to setups where a laser pulse is
scattered from electrons accelerated e.g. by a conventional linear accelerator
[6]. Furthermore, it can be used as an in-situ time-resolved diagnostic for
the laser-matter interaction [7]. However, despite these benefits, only a few
experimental investigations have been conducted on the subject using a purely
laser-based setup [8].

In this paper we describe an all-optical setup realizing Thomson
backscattering using a 2.4 $\mu$m thick aluminium foil target. In the course
of the discussion we first give a summary of the predicted properties of the
backscattered radiation relevant to this experiment. We then provide a brief
description of the electron acceleration from a thin foil target and the
evolution of the hot-electron sheath. This is followed by the discussion of
the optimization of the spatio-temporal overlap of the counterpropagating
pulses at the interaction region. Finally, we present the spectrum of the
generated XUV radiation.

###  1.1 Properties of the Thomson backscattered photons

The conversion of visible laser light into x-ray radiation through Thomson
backscattering involves the collision of the laser photons with relativistic
electrons which may have been generated from an intense laser pulse. In this
scenario the probe laser pulse and its wavelength appear length contracted by
the initial relativistic factor $\gamma_{0}=(1-\beta^{2}_{0})^{-1/2}$ of the
electron beam in its frame of reference, where $\beta_{0}=v_{0}/c$ is the
initial electron velocity (prior to the interaction with the probe pulse)
normalized to the speed of light $c$. Upon interaction, the electrons
oscillate in the field of the laser pulse leading to the emission of radiation
which is Doppler upshifted when measured in the laboratory frame [see Figure
1]. Consequently, the peak energy of the backscattered radiation in the
laboratory frame dominantly scales as $\gamma_{0}^{2}$ and is given by

$\hbar\omega_{\rm
x}=\frac{2\left(1+\cos\phi\right)\gamma^{2}_{0}}{1+a^{2}_{0}/2+\gamma^{2}_{0}\theta^{2}}n\hbar\omega_{0}$
(1)

where $\phi$ is the electron-probe beam interaction angle, _n_ is the harmonic
number of the emitted radiation, $\hbar$ is Planck’s constant,
$a_{0}=eE_{0}/(m_{\rm e}c\omega_{0}$) is the normalized vector potential of
the probe laser field, $e$ and $m_{\rm e}$ are the charge and rest mass of the
electron respectively, $E_{0}$ and $\omega_{0}$ denote the probe laser
electric field amplitude and frequency, respectively and $\theta$ is the angle
between the electron propagation direction and observation axis [9, 10]. The
$1+a^{2}_{0}/2$ factor accounts for the changes in the longitudinal velocity
of the electron as it traverses the probe pulse due to ponderomotive effects -
resulting in a shift of the peak to a lower energy in the intense field limit
($a_{0}^{2}\geq 1$) [11, 12]. Moreover, (1) implies that the upshift factor
varies from $2\gamma_{0}^{2}$ for transverse interactions ($\phi=90^{\circ}$)
to a maximum of $4\gamma_{0}^{2}$ for a head-on scattering geometry
($\phi=0^{\circ}$). It is also important to note that in (1) it is assumed
that the electrons are highly relativistic with $\beta_{0}\approx 1$ where the
radiation is mostly backscattered into small angles $\theta^{2}\ll 1$ [1].
This classical description of Thomson scattering is valid as long as the
incident photon energy is less than the electron rest mass in its rest frame
[13]. However, if the energy of the incident photon becomes comparable to the
electron energy, the electron recoil is no longer negligible and the
interaction can be described in the framework of Compton scattering [14]. The
scattered photon energy from this inelastic process is given by

$\hbar\omega_{\rm
x}=\frac{\left(1+\beta_{0}\cos\phi\right)}{1-\beta_{0}\cos\theta+\left(\hbar\omega_{0}/\gamma_{0}m_{\rm
e}c^{2}\right)\left(1-\cos\Delta\Theta\right)}\hbar\omega_{0}$ (2)

where $\cos\Delta\Theta=\cos\phi\cos\theta-\sin\phi\sin\theta\cos\varphi$ and
$\Delta\Theta$ is the angle between incident and scattered photons [15, 16].
Note that under the Thomson scattering condition $\gamma_{0}\hbar\omega_{0}\ll
m_{\rm e}c^{2}$ [13], (2) reduces to (1). Furthermore, the classical Thomson
formula is recovered by applying the Thomson scattering condition on (2) and
the limits $\phi=0^{\circ}$ and $\theta=0^{\circ}$ .

Figure 1: The interaction geometry of Thomson backscattering. An electron
propagating along z’ with a Lorentz factor $\gamma_{0}$ collides with a laser
pulse propagating in -z-direction emitting radiation at Doppler upconverted
frequencies. During the interaction, the z’ and z axes enclose the angle
$\phi$ [17].

The energy radiated by a single electron per frequency interval d$\omega$ and
per solid angle d$\Omega$ during the interaction can be computed from the
Lienard-Wiechert potentials [18]. For a head-on scattering geometry of a
relativistic electron with a linearly polarized probe laser pulse of arbitrary
intensity, the radiation along the backscattering direction
($\theta=0^{\circ}$) occurs only at the odd harmonics [1, 17]. Furthermore, in
the weak probe pulse limit ($a_{0}^{2}\ll 1$), significant radiation is only
generated at the fundamental ($n=1$). Higher-order odd harmonics start to
appear in the spectrum as $a_{0}$ approaches unity. The highest intensity,
however, is still observed at the fundamental. The energy distribution of the
backscattered radiation ($\theta=0^{\circ}$) at the fundamental harmonic
produced from the head-on scattering ($\phi=0^{\circ}$) of a single highly
relativistic electron ($\gamma_{0}\gg 1$) with a probe pulse for which
$a_{0}^{2}\leq 1$ is equal to

$\frac{\rmd^{2}I_{\rm{single}}}{\rmd\omega_{\rm x}\rmd\Omega}=16r_{\rm
e}m_{\rm
e}c\gamma^{2}N^{2}_{0}\frac{\alpha^{2}_{1}}{a^{2}_{0}}\left[1-\frac{\alpha_{1}}{2}\right]^{2}R(\omega_{\rm
x},\omega_{0})$ (3)

where $\alpha_{1}=a^{2}_{0}/[4\left(1+a^{2}_{0}/2\right)]$, $r_{\rm
e}=e^{2}/(4\pi\epsilon_{0}m_{\rm e}c^{2})$ is the classical electron radius,
$\epsilon_{0}$ is the vacuum permittivity, $N_{0}$ is the number of laser
oscillations encountered by the electron, and the resonance function
$R(\omega_{\rm x},\omega_{0})$ has the form

$R(\omega_{\rm x},\omega_{0})={\rm sinc}^{2}{(\omega_{\rm
x}-M_{0}\omega_{0})\bar{T}}$ (4)

where $\bar{T}=L_{0}/(2cM_{0})$, $M_{0}=4\gamma^{2}_{0}/(1+a^{2}_{0}/2)$ is
the relativistic Doppler upshift factor, and $L_{0}$ is the spatial laser
pulse length [1].

If not only a single electron is considered but a distribution which can be
described by an energy distribution $f(\gamma)$, the total fundamental
radiation spectrum is given by [8, 19]

$\frac{\rmd^{2}I_{\rm T}}{\rmd\omega_{\rm
x}\rmd\Omega}=\int\\!f(\gamma)\frac{\rmd^{2}I_{\rm{single}}}{\rmd\omega_{\rm
x}\rmd\Omega}\rmd\gamma.$ (5)

Integrated over the electron energy distribution, the spectrum is then
expressed as [7, 20]

$\frac{\rmd^{2}I_{\rm T}}{\rmd\omega_{\rm x}\rmd\Omega}=8r_{\rm e}m_{\rm
e}c\gamma^{3}N_{0}\frac{\alpha^{2}_{1}}{a^{2}_{0}}\left[1-\frac{\alpha_{1}}{2}\right]^{2}f(\gamma).$
(6)

The corresponding photon spectrum, which is calculated from $\Delta N_{\rm
x}/(\Delta\omega_{\rm x}\Delta\Omega)=(N_{\rm b}/\hbar\omega_{\rm
x})\rmd^{2}I_{\rm T}/(\rmd\omega_{\rm x}\rmd\Omega)$ where $N_{\rm b}$ is the
total number of electrons in the beam, is equal to

$\frac{\Delta N_{\rm x}}{\Delta\hbar\omega_{\rm
x}\Delta\Omega}=\frac{\alpha_{\rm
f}}{2\hbar\omega_{0}}N_{0}\gamma\alpha_{1}\left[1-\frac{\alpha_{1}}{2}\right]^{2}N_{\rm
b}f(\gamma).$ (7)

where $\alpha_{\rm f}=e^{2}/(4\pi\epsilon_{0}\hbar c)$ is the fine structure
constant [7]. The photons are emitted within a narrow cone of half-angle

$\Delta\theta_{1}\approx\frac{1}{\gamma_{0}}\sqrt{\frac{1+a^{2}_{0}/2}{N_{0}}}$
(8)

around $\theta=0^{\circ}$ [17].

Laser electron acceleration using solid targets and linear laser polarization
typically produces a broad energy distribution $N_{\rm b}f(\gamma)=\rmd N_{\rm
e}/\rmd\gamma$ where $\rmd N_{\rm e}/\rmd\gamma$ is the electron spectrum [8].
This can be approximately described by an exponentially decaying distribution
[21, 22],

$f(\gamma)=f_{0}\exp{\left[-\gamma m_{\rm e}c^{2}/k_{\rm B}T_{\rm e}\right]}$
(9)

Figure 2: Backscattered photon spectrum as a function of the longitudinal
electron temperature $T_{\rm e}$. The spectrum has been calculated assuming an
exponential electron energy distribution with a bunch charge of 30 nC, a laser
pulse with a temporal flat-top shape with a photon energy of 1.55 eV, 13
cycles pulse duration and $a_{0}=1.1$. The black dashed lines mark the
spectral range measured in the experiment. Figure 3: Photon spectrum as a
function of the observation angle for a mean electron energy of $k_{\rm
B}T_{\rm e}=1.1$ MeV and the same parameters as in Figure 2. The black dashed
lines mark the spectral range measured in the experiment

Here $\int\\!f(\gamma)\,\rmd\gamma=1$, $f_{0}$ is a normalization constant
[23], $k_{\rm B}$ is the Boltzmann constant and $T_{\rm e}$ the electron
quasi-temperature in the propagation direction. For illustration, (7) is used
to compute the backscattered photon spectrum emitted from an electron
distribution as described by (9). Figure 2 shows that at electron temperatures
$T_{\rm e}$ of less than about 0.8 MeV, the backscattered spectrum is peaked
at low photon energies. At higher electron temperatures, the emission extends
toward higher photon energies resulting in a broader and flatter spectrum of
the Thomson backscattered radiation.

The spatial distribution of the photon spectrum emitted by an exponential
electron distribution with a mean energy of $k_{\rm B}T_{\rm e}=1.1$ MeV is
shown in Figure 3. The emission peak is situated at $\theta=0$ rad and the
central radiation cone has a half-angle of $\Delta\theta_{1}\approx 0.12$ rad
as computed from (8). For larger observation angles, the spectrum is
restricted to lower photon energies.

###  1.2 The electron sheath

A relativistic electron population is generated by irradiating the front
surface of a thin foil target of thickness $d_{\rm t}$ at normal incidence
with an intense pump laser pulse of focal spot size $d_{\rm f}$. With this
configuration, the ponderomotive acceleration mechanism pushes the electron
population into and eventually through the target foil. Fast electrons exit at
the rear surface leaving a positively charged target behind. Due to the charge
separation, an electric field builds up. As a result, the majority of the
electrons of the distribution is forced to return to the target since they are
trapped in the induced electric field - forming an electron sheath [24].
Assuming a ballistic propagation of these electrons through the target, the
divergence half-angle of the beam is found to be $8^{\circ}$ 111The analytical
model proposed in [25] (which is supported by experimental results from [26,
27, 28]) predicts the dependence of the divergence angle of the electron beam
on the energy of the incident laser pulse. The divergence angle used in this
paper is consistent with a 1-J laser like the JETI laser system. [29]. From
geometrical estimates, the full-width at half maximum (FWHM) radial extent of
the electron sheath is computed as [28]

$w_{\rm ne}=d_{\rm f}+2d_{\rm t}\tan{8^{\circ}}.$ (10)

However, these electrons are pulled back by the arising field and are forced
to turn around towards the target [30]. This phenomenon causes a fountain-like
spreading of the electron sheath and hence is called the fountain effect. Once
these fountain electrons reach the target front side, they are reflected back
to the rear side then they come back again to the front surface and so on
[31]. This recirculation causes the electrons to lose energy and spread out
sidewards [32]. Consequently, a much larger radial extent of the electron
sheath than predicted by (10) is expected [24]. The electron sheath then
consists of a very dense center while the density rapidly drops in the outer
regions [32].

For the interaction of this electron sheath with a counterpropagating probe
pulse, it can be assumed that the scattering process is dominated by the dense
center of the electron sheath. And since the electron temperature is also
maximum at the sheath center [32], the contribution of fountain electrons to
the detected spectrum is minimal. In addition, the spatial distribution of the
backscattered radiation is centered in the electron’s forward direction i.e.
along z’ making an angle $\phi$ with respect to the z-axis. For the
fundamental harmonic, the emission is confined to a cone of half-angle
$\Delta\theta_{1}$. If the detection system (solid angle of
$\Delta\Omega=4\times 10^{-4}$sr with subtended half-angle of $\theta_{\rm
det}=0.01$ rad) is situated along the z-axis centered at $\phi=0$ rad, the
backscattered radiation from electrons with interaction angle
$\phi\geq\Delta\theta_{1}+\theta_{\rm det}$ is no longer measured. This
further minimizes the impact of fountain electrons to the Thomson
backscattered signal. Therefore, in this paper it is sufficient to consider
only the backscattered radiation from a head-on scattering geometry.

##  2 Experimental Setup

###  2.1 Layout

The experiment was carried out at the Jena Titanium:Sapphire TW laser system
(JETI). It provided linearly polarized pulses at a central wavelength of
$\lambda_{\mu}=0.8$ $\mu$m ($\hbar\omega_{0}=1.55$ eV) with $\tau=30$ fs
duration. A single pass plasma mirror was employed to improve the contrast
ratio between the main pulse and the pre-pulses (occurring 150 ps before the
main pulse) from $10^{-6}$ to $10^{-9}$ and the ASE background from $10^{-9}$
to $10^{-12}$ [33]. A diagram of the experimental setup is presented in Figure
4.

Figure 4: Experimental setup. The main JETI pulse was split into two by a 3-mm
thick dielectric-coated 90/10 beam splitter. Both pulses were then focused at
normal incidence onto the aluminium foil target using two f/2 $45^{\circ}$
off-axis parabolic (OAP) mirrors. The backscattered radiation was detected
using a XUV spectrometer and an AndorTM XUV CCD camera arrangement. Spatio-
temporal overlap between the two pulses was established using the shadowgram
of the interaction region and by the 0th-order image on the CCD camera.

The main pulse of the JETI laser, with beam diameter of $d_{\rm beam}=6.0$ cm,
was divided into a pump and a probe pulse by a 3-mm thick dielectric-coated
90/10 beam splitter. 90% of the pulse energy was utilized as the pump pulse
while the remaining 10% was employed as the probe pulse. Each of these pulses
was focused at normal incidence onto the target by two $45^{\circ}$ off-axis
gold-coated parabolic mirrors of effective focal length f$=12.0$ cm. Using
this configuration, the pump pulse was focused to a focal spot of
A${}_{\rm{Pump}}=8.9~{}\mu$m2 (FWHM) to a peak intensity of
$I_{\rm{Pump}}=1.9\times 10^{19}$ W/cm2 corresponding to a normalized vector
potential of $a^{\rm{Pump}}_{0}=3.0$. On the other hand, the probe pulse was
focused to a spot of A${}_{\rm{Probe}}=7.5~{}\mu$m2 (FWHM)222The spot size of
the pump pulse is slightly larger than the probe pulse due to its reflection
from the beam splitter. The probe pulse, in contrast, is transmitted.
Therefore, imperfections on the beam splitter surface do not significantly
alter the beam quality. to an intensity of $I_{\rm{Probe}}=2.8\times 10^{18}$
W/cm2 with $a^{\rm{Probe}}_{0}=1.1$.

The intense pump pulse accelerates electrons into the $d_{\rm t}=2.4~{}\mu$m
thick aluminium foil and the photons from the probe pulse impinging on the
target rear surface are backscattered off the electrons exiting the rear
surface. The backscattered radiation was collected through a 3-mm hole in the
off-axis parabolic mirror of the probe. A 0.5-T magnet located on axis at the
back of the parabolic mirror was then employed to prevent accelerated charged
particles from propagating through the detection system. An XUV spectrometer
employing a toroidal mirror with a collection solid angle of
$\Delta\Omega=4\times 10^{-4}$sr and a binary transmission grating made of
free-standing gold bars with 2000 lines/mm together with an AndorTM XUV CCD
camera make up the detection system in this experiment. Background low-energy
plasma emission as well as ambient radiation were blocked using two 200 nm
thick aluminium filters placed in front of the spectrometer. Additional lead
shielding was also placed at the entrance of the spectrometer to enhance the
suppression of background radiation. This gives an effective detection range
of 52 eV to 72.7 eV. The setup, excluding spectrometer and camera, was placed
in an octagonal vacuum chamber where a pressure of about $10^{-5}$ mbar was
applied.

###  2.2 Spatio-temporal Overlap

####  2.2.1 Alignment in the direction of the y, z and t axes

An additional temporally synchronized overlap probe was utilized to back-light
the pump-probe interaction region [see Figure 4]. It created a shadowgram of
the ionization front produced by each pulse individually when fired through
air. The shadowgram was imaged onto a CCD camera. The overlap in the y\- and
z\- axes of the laser foci was obtained by intersecting the region with the
peak plasma self-emission of each ionization front. To establish the spatial
overlap, the probe OAP focus was translated towards the pump OAP focus. Then
the corresponding temporal overlap was adjusted by compensating the delay
introduced during the probe OAP alignment with an appropriate translation of
the beam splitter. This was done by noting that shifting the beam splitter by
a distance of $\Delta l$ normal to its surface makes the pump pulse cover an
additional optical path of $\Delta\sigma=\sqrt{2}\Delta l$. Furthermore, the
delay stage of the overlap probe was scanned to mark the onset of each
ionization front. Then the position of the beam splitter was again adjusted to
balance the difference in the arrival time of both pulses. This was then used
to set the temporal delay of $\Delta t=0$ between the arrival of the two
pulses at the target position with a resolution of approximately 60 fs.

####  2.2.2 Alignment in the direction of the x and y axes

The alignment of the laser foci in direction of the x\- and y\- axes was
established by noting separately the pixel positions of the $0^{th}$ order
diffraction of the emitted light from each OAP as measured by the XUV CCD. The
optimum overlap of the foci with respect to these axes was then achieved by
matching these pixel positions. With this approach, the foci can be overlapped
with an accuracy of about $13~{}\mu$m in both transverse directions which
corresponds to half the pixel size.

####  2.2.3 Size of the interaction region

The radial extent of the electron sheath created by the pump laser pulse is
determined from (10) and is found to be $w_{\rm ne}=4~{}\mu$m. This
corresponds to a FWHM electron sheath area of $A_{\rm{Sheath}}=13~{}\mu$m2.
This matches well with the probe pulse focal spot
A${}_{\rm{Probe}}=7.5~{}\mu$m2 and is already sufficient for observing the
backscattering process. Moreover, due to the fountain effect an even larger
electron sheath spread is expected. In fact, Jäckel et al. observed that the
experimentally deduced radial extent is about a factor of 2.5 times larger
than the estimated value [24].

##  3 Experimental Results

Figure 5: Raw CCD image of the XUV spectrometer in false-colours taken during
a single laser shot. A logarithmic colour table was used for better image
rendering. To increase the spectral range accommodated by the spectrometer,
the grating was slightly rotated resulting in a minor tilt of the XUV spectral
trace. The aluminium filter edge at 72.7 eV in combination with the $0^{th}$
order diffraction were used to establish the pixel to wavelength calibration.

A typical raw image produced by the Thomson- scattered radiation from a single
laser shot is shown in Figure 5. The raw XUV spectrum, represented by the
slightly tilted and spectrally dispersed signal in the middle of the image, is
overlaid with single and multiple pixel events. A numerical filter was applied
to the image to remove isolated pixel events, thus separating the raw XUV
spectrum. Using the filter transmission and the response function of the
spectrometer-CCD system the number of photons is deduced from the CCD signal.
Given in Figure 6 is the XUV spectrum generated when the pump pulse arrived on
the target position 100 fs prior to the probe pulse.

Figure 6: XUV photon spectrum measured for a positive pump-probe delay of 100
fs. The error bars indicate the standard error using 16 subsequent laser
shots. The filter transmission (blue line) and the response function (green
dashed line) are included in the spectrum.

A direct evidence that Thomson backscattering is the dominant process in the
generation of high energy photons is the dependence of the XUV signal on the
pump-probe delay. Shown in Figure 7 is the summation of the number of counts
of the raw XUV spectra over the energy range between 52 eV and 72.7 eV for
different pump-probe delay. A negative delay indicates that the probe pulse
hits the target before the pump pulse while the opposite is true for a
positive delay. Figure 7 highlights the occurrence of a peak at 100 fs and the
gradual decrease in the count yield with increasing positive delay. At the
100-fs delay setting the detected XUV signal is enhanced by Thomson-
backscattered photons. This can be explained by the simultaneous presence of
the highest electron density and temperature at this instance in the
interaction area. At later times, the rear-surface electron sheath continually
expands longitudinally and in the radial direction [24]. This transverse
plasma expansion and the associated adiabatic cooling of the electron
population causes the steady decrease in the signal for increasing positive
delays. It is worth emphasizing that since the positive pump-probe delays have
femtosecond time-scales, the strong dependence of the XUV yield on the delay
can only be due to the Thomson backscattering process. Surprisingly, an
appreciable signal yield is also observed for negative pump-probe delays. In
this case the intense probe pulse generates electrons itself that are
accelerated in the opposite direction as compared to those originating from
the interaction of the target with the main pulse (i.e. they are moving from
the target rear side to the target front side). These electrons then form a
preplasma on the target front side into which the more intense pump pulse is
focused. At a delay of -400 fs, the preplasma created by the probe beam is
still hot. However, for a delay of -1.5 ps, the preplasma electrons will have
cooled down allowing an optimum energy coupling with the pump laser pulse.
Therefore the more intense pump pulse accelerates the electrons towards the
rear side of the target, generating a considerable signal from broadened line
emissions and Bremsstrahlung radiation [34, 35]. Indeed, Figure 7 shows an
elevated count yield at this delay.

Figure 7: Dependence of the integrated count yield of the XUV signal (energy
range between 52 eV and 72.7 eV) on the pump-probe delay. The error bars
indicate the standard error of the integrated count yield for 16 laser shots.
The red dashed line indicates the sum of the XUV count yield due to the pump
and the probe pulse.

Although the results above are consistent with Thomson backscattering, the
distribution of the backscattered signal can only be unambiguously determined
by isolating the background signal. In this particular case, however, it was
not straightforward to discriminate the exact background signal. This comes
from the fact that when both pulses interact with the target they essentially
share a common volume of the aluminium foil where the background and
backscattered signal are produced from. Whereas, when only either of the
pulses interacts with the foil target, the entire bunch of accelerated
electrons generates the background signal. It thus follows that the combined
signal yield of the pump and the probe is expected to be larger than the total
backscattered signal yield. The red dashed line in Figure 7 indicates the sum
of the signal yield of the pump and probe pulses. As predicted, it is found to
be higher than the signal obtained when both pulses interact with the target.

Figure 8:  The background-corrected XUV spectrum for a pump-probe delay of 100
fs showing statistically significant Thomson backscattered signal for photon
energies greater than 52 eV (black square). The background signal is estimated
from the average of the spectrum from -5 ps and +5 ps delays. The error bars
indicate the standard error from 16 laser shots. A theoretical plot of the XUV
spectrum obtained for an electron distribution with a mean kinetic energy of
$k_{\rm B}T_{\rm e}=1.1$ MeV (blue line) exhibits the same shape as the
background-corrected spectrum. The difference in the absolute photon yield
between the two plots can be explained by the imperfect overlap of the
electron beam with the laser pulse in the experiment.

An estimate of the background radiation can be found by using the average of
the signal from the -5-ps and +5-ps delays. The resulting XUV spectrum for 100
fs delay after background subtraction and a theoretical plot of the XUV
spectrum at an electron kinetic energy of 1.1 MeV are shown in Figure 8. The
backscattered spectrum is broad which obviously arises from the broad electron
energy distribution as noted in Figure 2. This is expected due to the
ponderomotive electron acceleration mechanism which is likely to be dominant
under our experimental conditions. Note that at normal incidence of the pump
pulse on the target, effects like resonance absorption or Brunel heating are
strongly suppressed [36]. It is also evident that Thomson-backscattered
photons are produced. However, the signal has statistical significance only at
energies greater than 52 eV. This is consistent with backscattering from the
escaping hot electron population: The ponderomotive scaling law predicts that
electrons oscillating in the field of an incident laser will gain a mean
kinetic energy of $k_{\rm B}T_{\rm e}=(\gamma_{0}-1)m_{\rm e}c^{2}$ where
$\gamma_{0}=\left(1+I_{\rm{Pump}}\lambda^{2}_{\mu}/1.37\times
10^{18}\textrm{W}\mu\textrm{m}^{2}/\textrm{cm}^{2}\right)^{1/2}$ [37].
Considering our pump pulse intensity, the mean kinetic energy of the electrons
correspond to $k_{\rm B}T_{\rm e}=1.1$ MeV with $\gamma_{0}=3.0$. A small
fraction of these hot electrons escape from the target rear surface charging
up the target to a potential of the order of $k_{\rm B}T_{\rm e}$ [38]. The
number of escaped electrons $N_{\rm esc}(t)$ and the self-capacitance of the
target $C_{\rm T}(t)$ then dictate the temporal evolution of the target
potential $V(t)$ such that $V(t)=N_{\rm esc}(t)e/C_{\rm T}(t)$ where $N_{\rm
esc}(t)=N_{\rm b}\exp(-eV(t)/k_{\rm B}T_{\rm e})$ and $C_{\rm T}(t)\approx
8\epsilon_{0}(r_{\rm f}+0.75ct)$ assuming that the charging expands from the
laser spot size of radius $r_{\rm f}$ with a radial velocity of the charge
wave approximated at $0.75c$ [38, 39, 40]. Consequently, once a sufficient
number of electrons have escaped, the bulk of the hot electron population
becomes electrostatically trapped on the target rear surface and only those
that are energetic enough to overcome the developing potential barrier can
escape [41]. Therefore, backscattering from the escaping hot electron
component having $\gamma_{0}\geq 3.0$ will produce photons at the Doppler
upshifted energies of $\hbar\omega_{\rm x}\geq$ 56 eV. Furthermore, since the
escaping hot electrons scatter more strongly than their lower energy
counterparts, they contribute dominantly to the backscattered spectrum. The
consistency between the experimentally observed behaviour of the XUV spectrum
and theoretical predictions indicate the presence of Thomson backscattering in
the photon production.

##  4 Conclusion

An all-optical, laser-based setup was presented for investigating Thomson
backscattering. At the present conditions, the photon spectrum produced is
broad due to the broad electron distribution. A statistically significant
photon yield is detected at photon energies greater than 52 eV which
corresponds to backscattering from the escaping hot electron population. In
addition, the XUV signal exhibits a strong dependence on the difference
between the arrival time of the pump and the probe pulses at the target
position. A maximum XUV signal is obtained when the pump pulse interacts with
the target 100 fs before the probe pulse. The XUV signal also decreases with
increasing positive pump-probe delays. The behaviour of the XUV spectra,
particularly at increasing positive delays clearly indicate that the source of
these high-energy photons is Thomson backscattering. It was concluded that the
highest flux and energy in the Thomson backscattered photon signal is
generated by the rear-surface electron sheath having both the highest density
and also temperature. Since these parameters are also of great importance for
the acceleration of ions from the rear surface of thin foils [29], the
spectral and temporal behaviour of the Thomson backscattered signal has the
potential to provide additional information about the temporal evolution of
the ion-acceleration fields.

Optimization of the experiment will involve the use of electron sheets with a
quasi-monoenergetic distribution. Using ultra-thin diamond-like carbon foils
being accelerated in the radiation pressure acceleration regime might be a
future alternative to produce both monoenergetic electron and - as a
consequence - XUV and even x-ray spectra.

This work was supported by the Deutsche Forschungsgemeinschaft (SFB/TR 18) and
the German Federal Ministry of Education and Research (BMBF) under contract
05K10SJ2. C.R. acknowledges the support from the Carl Zeiss Foundation. We
acknowledge the contributions of Burgard Beleites and Falk Ronneberger to this
work by maintaining the JETI laser system.

## References

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