All-optical measurement of the hot electron sheath driving laser ion acceleration from thin foils

We present experimental results from an all-optical diagnostic method to directly measure the evolution of the hot-electron distribution driving the acceleration of ions from thin foils using high-intensity lasers. Central parameters of laser ion acceleration such as the hot-electron density, the temperature distribution and the conversion efficiency from laser pulse energy into hot electrons become comprehensively accessible with this technique.


Introduction
In recent years, high-intensity laser systems have proven to be promising candidates for nextgeneration particle accelerators [1,2]. Laser-generated, relativistic plasmas can provide electric fields that are by several orders of magnitude stronger than the fields used in conventional accelerators and thus significantly reduce the particle acceleration length. In addition, it has been demonstrated that electron [3]- [7], proton [8,9] and ion [10] pulses with monoenergetic features of a few per cent spectral bandwidth can be reliably generated, paving the way for possible applications (e.g. [11,12]). Other unique properties of this source of energetic particles encompass ultrashort pulse duration [13]- [15] and excellent transverse and longitudinal emittances at the source [16] that allow for good focusability of the particle beams.
Until now, most improvements concerning particle properties have been achieved by improving the laser and target parameters. For example, enhancement of the maximum ion energy or spectral manipulation can be achieved by employing special target designs using microstructured [8,9,17] or nanolayered thin foils [10,18], ultrathin foils [19]- [21] and droplet targets [22,23] or by employing subsequent beam shaping devices such as fs-gated, electrostatic focusing optics as a so-called laser-driven microlens [24] or cascaded acceleration setups [25]. All these approaches follow theoretical models about the nature of the underlying physical processes. Although these models were continuously refined in accordance with novel experimental findings, no adequate real-time observation tool has been available for studying the interaction and thus the acceleration itself. So far, proton deflectometry is the only diagnostic capable of resolving the relevant physical processes leading to laser ion acceleration by measuring the electric fields within the electron sheath [26]- [30]. However, proton deflectometry is constrained by picosecond temporal resolution, insufficient for tracking the rapid, femtosecond-scale laser plasma dynamics responsible for particle acceleration, and requires a complicated setup of two correlated laser accelerators to probe the generation of one particle beam with the other. This mismatch has repeatedly led to calls for improvement of diagnostics to better understand the nature of laser particle acceleration.
Here we present an all-optical method for directly probing and reconstructing the electron distribution that drives laser ion acceleration from thin foils in real time. The use of an optical probe pulse of 100 fs pulse duration offers an unprecedented temporal resolution operating on 3 the theoretically predicted time scale of the acceleration process determined by the duration of the driving laser pulse [14,29].
A laser pulse (with intensity I L ) impinges on the front surface of a thin foil of µm thickness and generates-mainly via ponderomotive acceleration-a hot-electron component that propagates through the target foil. The majority of these electrons are trapped in the electric potential φ arising from the charge separation. Thus, the hot electrons can be described by a Boltzmann distribution n e (z) = n e0 exp {eφ(z)/k B T e }, where the hot-electron temperature k B T e = P is determined by using the ponderomotive potential P = m e c 2 ((1 + I L λ 2 L /1.37 × 10 18 W cm −2 µm 2 ) 1/2 − 1) of the laser pulse, with I L and λ L being the laser intensity and wavelength, respectively. n e0 is the hot-electron density inside the target foil and e is the elementary charge.
Under these conditions, the electron density n e follows an exponentially decaying density distribution in the longitudinal direction (normal to the target rear surface) behind the target foil related to the Debye length λ D = ( 0 k B T e /e 2 n e ) 1/2 . For t = 0, an analytical solution of the Poisson equation can be derived outside the target foil (n i0 (z >0) = 0) for the one-dimensional (1D) case [36]: where e N is Euler's number.
In simple estimations, the radial extent w n e (full-width at half-maximum (FWHM)) of this electron sheath has so far been calculated using the focal spot size on the target front surface, d foc (FWHM), while assuming a ballistic propagation of the hot electrons through the target. For a target thickness of d t and an electron half-opening angle of θ, this leads to w n e = (d foc + 2d t tan θ ) [37]. The electron propagation through the target can be assumed to be symmetric around the target normal direction for a sufficiently short front-surface plasma scale length even for oblique incidence of the driving laser pulse on the target front surface [38,39]. Previous measurements of the contrast ratio of the laser system allow us to conclude that the front-surface scale length is shorter than λ L .
In contrast to the transverse diameter of the rear-surface electron sheath estimated from the laser focal spot size, a significantly larger diameter can be expected because of effects such as electron recirculation [40] or the fountain effect [41]- [44] that are not included in the above-mentioned simple estimations, leading to a larger source size of the accelerated ions [45]. However, in the case of normal electron propagation through the target, it can be expected that the rear-surface electron sheath still preserves its cylindrical symmetry with respect to the target normal. For sufficiently large aspect ratios (i.e. the ratio of the extent of the electron sheath in the target normal direction and the lateral diameter of the sheath), the generated electric fields can still be approximated using the above-mentioned 1D model.
Within the cloud of hot electrons at the rear surface a strong electric field E TNSA = − ∂φ(z)/∂z| z=0 = √ 2/e N √ k B T e n e / 0 is generated, which ionizes atoms at the target rear surface immediately [46]. Once these ions have been generated they are accelerated normally to the target surface up to MeV energies. As the acceleration is sensitive to the charge to mass ratio q/m, protons are accelerated most effectively. The electric field in the sheath is mainly determined by the distributions of number density and temperature of the electrons. Although it is likely that the total electron distribution is a combination of more than one temperature component, only the hottest-electron component can leak out of the target rear surface over distances of 1 µm or higher [41,47] at the beginning of the ion acceleration process. While at later times the less energetic electrons can also leave the target, the density distribution of the ions still exhibits a very steep, almost step-like density distribution at these early times. Due to their Debye length in the nm range, the extent of the low-temperature electron component at the target rear surface is only tens of nanometers and this is not resolvable with the present setup. Although the low-temperature component may increase the electric field strength close to the target rear surface, the electric field in the electron cloud is determined only by the hot-electron component.
Typical estimations following the geometrical assumptions concerning the electron sheath extent with an electron beam divergence half-angle of θ ≈ 8 • [48] as well as an energy conversion of laser light into hot electrons of 10% lead to n e0 = 9.4 × 10 19 cm −3 [14,37]. Together with I L = 0.94 × 10 19 W cm −2 , one expects that k B T e = 0.7 MeV, λ D = 0.6 µm and w n e = 8.6 µm for the conditions of our experiment. Consequently, the ionizing and accelerating electric field strength is 1.1 TV m −1 . To test the validity of these estimations experimentally, a diagnostic sensitive to the electron distribution with high spatial (of the order of microns) and temporal resolution (of the order of the laser pulse duration) is necessary.

Experimental setup
The experiments have been carried out at the Jena 10 TW titanium:sapphire laser system Jeti, which delivered pulses of E L = 500 mJ within τ L = 80 fs (FWHM) at a central wavelength of λ L = 800 nm. An f /2 off-axis parabolic gold mirror focused the laser pulses into a focal spot of A L = 11.5 µm 2 (FWHM), which contained 30% of the energy, leading to intensities of I L = 1.1 × 10 19 W cm −2 on the target taking into account the incidence angle of 45 • . The probe beam was separated from the compressed main pulse, telescoped down to a diameter of 1 mm and frequency-doubled using a BBO crystal. Due to the additionally accumulated optical path length, the final pulse duration of the probe pulse on the target was τ 2ω ≈ 100 fs owing to group velocity dispersion. A variable delay line with a minimum step size of 66 fs allowed us to choose different probing times within a time window of 4 ns around the arrival of the peak of the main pulse on the target front surface. Figure 1 shows a schematic overview of the setup. Titanium foils of d t = 6 µm thickness were mounted on specially bent target frames (radius of curvature 5 cm). The 2ω probe pulse passed the curved rear surface of the target tangentially and collected a phase shift ≈ ω L /(2cn c ) n e ds (n e n c ) during its propagation through the electron sheath. Here ω L and c are the laser frequency and the speed of light, respectively; n c = 0 m e ω 2 L /e 2 is the critical density. A Nomarski interferometer comprising a Wollaston prism (separation angle 1 • ) and a polarizer imaged the region of interest onto a 12 -bit CCD camera using a high-quality f /2 lens. The 1.1 µm spatial resolution of this setup was found to be close to the diffraction limit using a standard resolution test pattern. The evaluation of the interferograms was performed using IDEA software [49]. Assuming cylindrical symmetry of the rear-surface plasma with respect to laser pulse I = 1.1 x 10 W/cm 19 2 bent thin foil target the target normal direction, the absolute electron density distribution could be obtained from the phase shift plots via an Abel inversion. The ellipticity of the rear-surface sheath was confirmed to be smaller than 1:1.3 by a different setup using reflectivity probing of the rear surface. The target foils were 25 × 25 mm 2 in size. A custom-designed translation and rotation stage arrangement allowed us to take up to 40 shots onto each foil without breaking the vacuum. A robust alignment procedure guaranteed that the main laser pulse hit the foil exactly at the osculation point of the tangential probe beam. The accuracy of the alignment corresponds to a maximum shaded region of 0.1 µm thickness in the target normal direction, which might not have been accessible by our imaging system.
In order to determine the proton energy, a CR39 nuclear track detection plastic positioned 15 cm behind the target foil on the target normal axis was covered with aluminum filter strips of increasing thickness towards the beam center. Figure 2 shows the density distribution of the rear-surface electron sheath recorded at t = 0. The time t = 0 refers to the first moment when a signal can be detected on the target rear surface after successively moving the delay stage towards earlier times in steps of 66 fs. The electron distribution shows an exponential decay with a half-value length of 1.5 µm from n e = 3 × 10 19 to 0.5 × 10 19 cm −3 over a distance of 7.5 µm in the target normal direction as well as a Gaussian shape in the radial direction with 21 µm FWHM.

Properties of the initial electron density distribution
Applying equation (1), the density distribution was approximated by an exponential drop in the longitudinal direction described by a Debye length of λ D = (1.0 ± 0.2) µm and a hotelectron density inside the target of n e0 = (8.4 ± 0.4) × 10 19 cm −3 directly from the measured data. The errors have been determined by analyzing different shots taken under the same experimental conditions, by determining the influence of parameters that are necessary to initialize the Abel inversion and, finally, by the deviation of the deduced values from the theoretical fit curve that is given by equation (1).
The density profile is in good agreement with the predictions of the theoretical model. The experimentally determined electron density and scale length differ slightly from the theoretical values in terms of absolute numbers: n e0 is overestimated by 12%, leading to a 40% smaller predicted value of λ D . The same is true for the electron temperature, which was deduced to be k B T e = (1.5 ± 0.4) MeV, in contrast to the value given by the ponderomotive potential of the laser which is smaller by a factor of 2. The initial electric field strength derived from our measurements of E TNSA = (1.1 ± 0.4) TV m −1 is found to be in good agreement with the theoretical model as the effects from the higher temperature and the smaller Debye length tend to compensate each other.
Note that the experimentally measured initial distribution of n e can, to first order, be approximated to be 1D, i.e. its decay in the z direction is much faster than that in the radial direction with an aspect ratio of 1:14 given by the half-values, which justifies the application of the analytical solutions of the 1D model [36,50].

Efficiency of energy conversion
A parameter of central interest for understanding the interaction process is the energy conversion efficiency η of laser light into hot electrons: η = N e k B T e /E L . The values from the literature vary between a few per cent and several ten per cent, and were also predicted to follow an intensity scaling η = 1.2 × 10 −15 I 0.75 L [14,51,52]. From our measurements, we can deduce the energy of the electrons in the rear-surface sheath E Sheath directly by integrating the number density over space to obtain the total number of sheath electrons N Sheath = (5.1 ± 0.1) × 10 10 and multiplying it by their deduced average temperature k B T e = (1.5 ± 0.4) MeV.
This leads to E Sheath = (12 ± 4) mJ, which would correspond to a conversion efficiency of η Sheath = (2.6 ± 0.8)%. The total conversion efficiency of laser light into hot electrons can be deduced by the following estimation. We assume that, after being accelerated by the laser, the hot electrons occupy a cylindrical volume V inside the target of V = d t π(w n e /2) 2 , where d t and w n e are the target thickness and the radial extent of the rear surface sheath, respectively. Multiplying this volume by the hot electron density inside the target n e0 and their temperature k B T e that we deduced from our measurements yields a total energy of the hot electrons of E hot = (42 ± 13) mJ. Using this estimated number, the conversion efficiency of laser energy into hot electrons is η = (9.1±2.8)%. This value is in good agreement with the literature; however, it does not support the intensity scaling η(I L = 1.1 × 10 19 W cm −2 ) = 23% as mentioned in [14,51,52]. Note that for the determination of this value we took into account only that part of the electron population exhibiting the highest quasi-temperature.
A comparison with 3D-PIC (particle-in-cell) simulations by Pukhov [41] supports our experimental results strongly. For comparable laser and target conditions (I L,sim = 10 19 W cm −2 , d t,sim = 12 µm), an electron sheath of 20 µm × 6 µm radial to longitudinal extent is found, strikingly similar to our experimental results. The simulation yields a slightly higher initial electron density of a few 10 20 cm −3 , which might be ascribed to differences in the electron transport or the conversion efficiency of laser light into hot electrons assumed for the numerical modelling.
Since the numerical simulation results also support the result of our measurement of the radial extent of the electron density distribution w n e = 21 µm in contrast to the simple estimation for w n e = 8.6 µm due to electrons propagating ballistically through the target, this might point to an experimental observation of the so-called fountain effect [41]- [44]. A magnetically collimated electron beam leaving the back of the target spreads and is drawn back towards the surface and this causes a rapid broadening of the electron density distribution.

Duration of ion acceleration
Differently filtered CR39 plates show proton energies larger than 2.9 MeV, which is also supported by previous measurements with an ion spectrometer [8,9] in a similar setup, where proton cutoff energies of 4 MeV were measured. This observed maximum energy can be used to estimate the effective acceleration time of the protons in the TNSA sheath field-a parameter of central importance to many theoretical models. Figure 3 shows a plot of the maximum proton energy acquired by a proton propagating through the sheath field caused by the electron distribution of figure 2 as a black line, which was obtained by solving the 1D equation of motion numerically. This plot is contrasted with the plasma expansion model displayed as a gray line [50], again using the experimental results of electron temperature and density. For early acceleration times we find very good agreement between the two models. However, after approximately 300 fs, a discrepancy occurs due to negligence of the ion front expansion in the simple quasi-static 1D approach.
The measured maximum proton energy of E max 2.9 MeV corresponds to an acceleration time of τ acc,meas 280 fs, which is in excellent accord with estimates by Fuchs et al τ acc 2.5 (τ L + 60 fs) ≈ 350 fs based on a refined theoretical model and a proton energy scan for different laser parameters [29]. Our findings confirm this long-standing model by using a direct experimental probing technique.

Temporal evolution of the electron sheath
Varying the delay of the probe pulse with respect to the main pulse, the temporal evolution of the electron sheath has been studied as shown in figure 4  Since the analytical description of the longitudinal decay of n e (z) is valid only for t = 0, we assume the self-similar expansion of a hydrogen plasma into vacuum in order to model our observations. We further assume that the electron density profile expands with the ion sound speedż(t, n e = const) = c s (t) = (Z k B T e (t)/m i ) 1/2 and, hence, contains information on the momentary hot-electron temperature. The results of this approach are shown in figure 5. Its relative uncertainty was determined to be less than 30%.
The expansion velocity is found to decrease with time, indicating a deceleration of the plasma expansion. This behavior is expected owing to cooling of the hot-electron population driving the expansion. In particular, the rapid decrease of the electron temperature in figure 5(b) indicates the adiabatic cooling behavior expected from an expanding plasma [53]. For later times, we find that k B T e ∼ t −(1.4±0.2) as shown by the gray line. This exponential fit is in accord with numerical simulations by Mora [53] and falls into the transition between the limits of ultrarelativistic (k B T e ∼ t −1 ) and nonrelativistic (k B T e ∼ t −2 ) electron description. Thus, it strongly supports the description of the TNSA process as an adiabatic expansion of a plasma at the rear surface of a thin foil at later times.
For the hot-electron temperature, relativistic values were found for the first three time steps up to 400 fs only. From this upper limit for the duration of a relativistic interaction, one can make further predictions about the acceleration time period that can be narrowed down further to 280 fs τ acc 400 fs in accord with the predictions in [29].
In figure 5(c), the energy content of each electron density plot is evaluated over time assuming the above-calculated hot-electron temperature. We find that the energy contained by the sheath peaks at 200 fs, corresponding to a slightly higher energy portion of η Sheath = (4.1 ± 1.2)% at that time than reported above. This value marks an upper boundary for the energy transfer into the ions, which is in good agreement with table 1 of [54].

Conclusion
We have presented the experimental results from an all-optical scheme for the direct measurement of the electron distribution driving the laser ion acceleration from thin foils. The use of a synchronized probe beam and a special target geometry enabled temporal and spatial resolution a factor of 10 better than previous measurements made with proton deflectometry or ps laser pulses. This setup allows direct deduction of the Debye length λ D = (1.0 ± 0.2) µm and the undisturbed electron density inside the target n e0 = (8.4 ± 0.4) × 10 19 cm −3 , as well as calculation of the hot-electron temperature k B T e = (1.5 ± 0.4) MeV at the point in time when the acceleration process starts. The results are strongly supported by 3D-PIC simulations [41] and confirm the acceleration time prediction of Fuchs et al [29] regarding the maximum proton energy. Finally, a time-resolved measurement of the electron sheath highlighted the adiabatic nature of the cooling of the electron population and allowed determination of the energy conversion of laser light into hot electrons of the sheath to be η Sheath = (4.1 ± 1.2)%. The total conversion efficiency into hot electrons has been estimated as η = (9.1 ± 2.8)%. The proposed setup is, in principle, transferable to any other laser ion acceleration experiment under the stipulation that the targets can be bent or the rear surface can be made accessible in any other way, which is the case for most of the proposed target improvements.
Making direct, quantitative measurements of the electron sheath that drives ion acceleration from thin foils adds significantly to the understanding of laser ion acceleration and the underlying physics. In the future, it might help in improving the parameters of the generated particle pulses, which is a prerequisite for a large number of envisaged applications.