ACTA PHYSICA POLONICA A No. 4 Vol. 138 (2020)

Special Issue of the International Conference on Research and Applications of Plasmas (PLASMA-2019)

Target Charging, Strong Electromagnetic Pulse Emissionand Proton Acceleration from Thin Foils

at 10 TW IPPLM Femtosecond Laser Facility

P. Rączkaa,∗, J.-L. Duboisb, S. Hulinc, M. Rosińskia, L. Ryća,P. Parysa, A. Zaraś-Szydłowskaa, D. Terwińskaa, P. Tchórza,J. Badziaka, J. Ribolzic, V. Tikhonchukb,d and J. Wołowskia

aInstitute of Plasma Physics and Laser Microfusion, Hery 23, 01-497 Warsaw, PolandbCELIA, University of Bordeaux-CNRS-CEA, Talence, FrancecCEA-CESTA, Le Barp, FrancedELI-Beamlines, Inst. of Physics, Czech Academy of Sciences, Dolní Břežany, Czech Republic

Doi: 10.12693/APhysPolA.138.593 ∗e-mail: piotr.raczka@ifpilm.pl

It is well known that the laser–target interaction at high power and high laser intensity may resultin the emission of strong electromagnetic pulses with frequencies in the range from tens of MHz toa few GHz and the duration of hundreds of nanoseconds. It was recently pointed out that the electricpolarization of the target and the resulting neutralization current play an important role in the electro-magnetic pulses emission. The target charge and the neutralization current generated with laser pulsesof 30 fs to several ps duration were studied in detail at the Eclipse laser facility at CELIA, Bordeaux,with the laser pulse energy at the level of up to 100 mJ on target. In this contribution we report onthe measurements of the target charge and the electromagnetic pulses performed at the 10 TW fs laserfacility in IPPLM, Warsaw, for the laser pulse energies reaching 400 mJ on target and the laser pulseduration in the range of 50 fs to 400 fs. Due to higher laser pulse energy, the effect of proton accelera-tion off the rear side of the thin foil targets could be easily observed. This is an important bonus sincethe mechanism of target normal sheath acceleration of protons which prevails in these laser conditionsis closely related to the mechanism of charge ejection from the target, thus providing an additionalconstraint on any attempts at modeling of the target charge. The collected data is confronted withsimplified models of charge ejection and target normal sheath acceleration of protons and qualitativeagreement is found.

topics: laser–plasma interaction, high intensity lasers, electromagnetic pulses, laser-driven ionacceleration

1. Introduction

There has recently been an increased interest inthe study of strong electromagnetic pulses (EMP)emitted as a result of laser–target interaction athigh power and high intensity laser facilities [1–18].Such pulses have frequencies in the range fromtens of MHz to a few GHz and duration of hun-dreds of nanoseconds. Besides being an interest-ing physical phenomenon, EMPs are also of con-siderable practical concern in the forthcoming eraof multi-PW laser facilities because they may af-fect data collection and disturb electronic devicesused in experiments. It has recently been em-phasized that the charge deposition on target asa result of laser–target interaction and the en-suing neutralization current play a major role inthe EMP emission [3, 6]. Measurements of the neu-tralization current and the target charge were per-formed at the Eclipse laser facility at CELIA,Bordeaux, dealing with thick to moderately thin

targets [3, 6, 17]. The interest of the IPPLM groupis in the target charge deposition on ultrathin (µmscale) targets which are commonly used in ex-periments involving laser-driven ion acceleration.An experiment with such targets was performedat the Eclipse laser facility in collaboration withthe CELIA group, with maximum laser energies ontarget at the level of 100 mJ [14, 15].

In this paper, we report on the measurement ofthe target charge and the EMP signal at the 10 TWfemtosecond laser facility at the IPPLM, Warsaw,for laser pulse energies reaching 400 mJ on target.Laser pulse energies at this level resulted in clearlyobservable laser acceleration of protons. The pro-cess of laser acceleration of protons in this regimerelies on the same physical mechanisms that areresponsible for the charge deposition on the tar-get. Thus, recording the target charge in correla-tion with the energies of laser accelerated protons,we obtain a data set that allows for a stringent test

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of target charge deposition models. In the follow-ing sections, we describe the experimental setup andpresent preliminary results on the charge depositedon the target as a function of laser pulse energyat fixed pulse duration and laser pulse duration atfixed laser energy. We also present maximum pro-ton energies as a function of the laser pulse energyfor fixed pulse duration. We confront this data withtwo simple models of charge deposition and protonacceleration. Finally, we discuss the electromag-netic field of the strong pulse generated by the laser–target interaction. Some results on the EMP miti-gation which were obtained in this experiment werereported in a previous publication [18].

2. Experimental setup

The experiment was conducted at the 10 TWTi:sapphire laser facility at IPPLM, Warsaw, whichdelivers a beam with 810 nm central wavelengthand quite good intensity contrast of 5 × 10−9. Thebeam was focused by an off-axis parabolic mirrorto a spot of approximately 12 µm full width at halfmaximum (FWHM) and was incident on the targetat the angle of 5 (i.e., the target was rotated rel-ative to the laser beam axis). In this experiment,the laser energy on target was varied between 130and 400 mJ and the pulse duration was varied be-tween 49 and 320 fs. The laser pulse energy wasmeasured in each shot using an auxiliary beam leakafter the compressor. The uncertainty in the pulseenergy is dominated by about 5% systematic uncer-tainty in the calibration procedure that is requiredto correlate the energy of the leak with the energythat is actually deposited on the target. The uncer-tainty in the pulse duration varied between 0.3 fsfor 49 fs pulses to 13 fs for the longest pulses, re-sulting from the uncertainty in the pulse duration asa function of the compressor setting. For the highestlaser parameters in this experiment — 400 mJ totalenergy on target and 49 fs pulse duration — the es-timated average intensity over the FWHM spot is2.7 × 1018 W/cm2, assuming that spatial and tem-poral beam profiles are Gaussian.

The targets in this experiment were custom-made, with 6 µm Al foils pasted over 1 mm holesdrilled in a Cu “pill” 1 mm thick and 10 mm in di-ameter, which in turn was placed in the “lollipop”target holder used in the preceding experiment atthe Eclipse laser [14]. The surface of thin foil tar-gets prepared in natural environment is always con-taminated with hydrocarbon and water moleculeswhich form an ultrathin (nm scale) layer that be-comes the source of protons emitted from the foilupon irradiation by a high-intensity laser. The tar-get holder had the form of a brass ring 14 mm indiameter placed on a thin brass wire 30 mm long.It was directly connected via a 50 Ω SMA mount toa coaxial cable which allowed for a direct measure-ment of the target neutralization current. The SMAmount was attached to an electrically grounded thin

Al plate 160 × 160 mm2 to ensure the dominantdipole character of the electromagnetic emission ofthe target holder acting as an antenna.

To monitor the electromagnetic pulses insidethe experimental chamber, several probes were usedin this experiment but in this paper we only re-port results obtained with a commercial ProdynRB230 B-dot probe, coupled with the Prodyn BIB-100G balun. The RB230 probe was placed 41 mmabove the target, 218 mm from the target alongthe incident laser beam direction and 91 mm tothe right from the laser beam in the horizontal planeand was sensitive to the ortho-radial component ofthe magnetic induction field relative to the verti-cal axis. This component is expected to be domi-nant if the main driving process of the electromag-netic pulse emission is the neutralization current inthe (vertical) target stalk. Experience from pre-vious experiments for the laser and target param-eters similar to ours [3, 6] has shown that laser–target interaction results in a single strong shortspike of the return current which generates a sin-gle strong electromagnetic pulse that expands intothe experimental chamber. This pulse is reflectedfrom the chamber walls and the equipment placedinside the chamber, resulting in decaying electro-magnetic oscillations inside the chamber lasting forhundreds of nanoseconds. The shape and magni-tude of the strong initial pulse depends on the typeof the target and the form of the target supportbut is independent of the form of the experimentalchamber and the distribution of diagnostic equip-ment inside the chamber. Characteristics of thisstrong initial pulse are important from the point ofview of potential damage to the electric and elec-tronic equipment inside and outside the chamber.On the other hand, the duration and the spec-tral composition of the decaying oscillations de-pends strongly on the size and shape of the exper-imental chamber and on the amount of diagnosticequipment placed inside it. The information aboutthe decaying tail of electromagnetic oscillations isimportant for reduction of electromagnetic interfer-ence in the data collection process; for example, byenlarging the flight path of the recorded ions wemay reduce or even completely eliminate the effectof EMP on the readings of solid state ion detec-tors. Both pieces of information on the EMP sig-nal — the characteristics of the short strong initialpulse and the duration and spectrum of the decay-ing oscillations — may be meaningfully extractedfrom the data of even a single electromagnetic probeplaced inside the chamber.

To monitor the laser-accelerated ions, a silicondetector was used [19], denoted as FLM. This de-tector was mounted at a distance of 1863 mmfrom the target at the end of a long tube protrud-ing from the experimental chamber exactly alongthe direction of the incidence of the laser beam.A schematic top view of the experimental chamberis shown in Fig. 1.

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Fig. 1. A schematic view of the experimentalsetup at the 10 TW IPPLM laser facility. Themounting of the FLM ion detector is not shown inscale — it was actually mounted on a long tubeprotruding from the experimental chamber.

3. Measurement of the chargedeposited on target

The total charge deposited on target is extractedfrom the data on the target neutralization current.The typical neutralization current profile is shownin Fig. 2. It has the form of a narrow spike fol-lowed by a few rapidly decaying oscillations, withthe basic pulsation lasting no longer than 1 ns.The same form was observed in almost all the shots,with a difference only in the height of the initialspike. This form is almost identical to what wasobserved in earlier experiments at lower laser ener-gies [3, 14] with a small but interesting difference,namely, the presence of a noticeable negative spikejust before the main (positive) spike.

By integrating the neutralization current we ob-tain the target charge as a function of time, fromwhich we can extract the charge deposited in laser–target interaction by taking a difference of q(t) be-fore and after the main spike (Fig. 3). The chargestate after the main spike is estimated as an averageof the target charge at the first maximum and firstminimum after the main spike. Other prescriptionsare possible but it turns out they generate resultswithin ±1 nC.

The target charge as a function of the laser pulseenergy for fixed pulse duration ≈ 50 fs is shownin Fig. 4. The central line of the linear best fit,namely, Q = (0.077 ± 0.005)EL + (−5 ± 2) isshown as well to guide the eye. We find that sim-ilarly to previous experiments [3, 6, 14], the targetcharge displays approximately a linear dependenceon the laser pulse energy. The data shows somescatter, particularly for a number of data pointsobtained for energies very close to 390 mJ, i.e.,at the top of the considered laser energy range.Our understanding is that rather than being a man-ifestation of some complicated physical processesthis is a result of random factors, such as inherent

Fig. 2. The temporal dependence of the neutral-ization current in shot #43 at the laser pulse energy389 mJ and the laser pulse duration 49 fs.

Fig. 3. As in Fig. 2 but for the dependence ofthe target charge.

fluctuations in the performance of a high-powerlaser, variations in the target shape, which are un-avoidable in custom-made targets, and human in-accuracy, since for each shot the target is broughtinto focus by manual adjustments. This view issupported by the analysis of the maximum ener-gies of the accelerated protons, discussed furtherbelow. It should be noted that despite the fact thatthe pulse energies involved in this experiment arelarger by a factor of 4, as compared to the previousexperiments, the values of the target charge are infact of similar magnitude. This could be explainedby the fact that the laser contrast in this experimentis much better than in the previous experiments andhence there is much less preplasma present on tar-get at the arrival of the main pulse. It was notedin [14] that the target charge appears to be verysensitive to the presence of a preplasma.

The target charge as a function of the laserpulse duration at fixed laser pulse energy is shownin Fig. 5. We find that the dependence is muchweaker than it might be expected on the pure scal-ing of the laser intensity which is consistent with

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Fig. 4. The target charge as a function of the laserpulse energy, at laser pulse duration ≈ 50 fs,compared to predictions of the model by Poyéet al. [17], for two values of the laser ab-sorption coefficient η = 0.10, 0.12. Also shown isthe line indicating the best linear fit to the data:Q = (0.077± 0.005)EL + (−5± 2).

Fig. 5. The target charge as a function of the laserpulse duration at fixed laser pulse energy. The laserenergy on target was approximately 390 mJ. Forcomparison, we show predictions of the model byPoyé et al. [17], for two values of the laser absorptioncoefficient η = 0.10, 0.12.

what was seen in the previous experiments [3, 6, 14].It should be noted that it is the laser energy be-fore the compressor that was being kept fixed; somefluctuations in the pulse energy after the compres-sor were observed in this part of the experimentbut the data displayed in Fig. 5 is not corrected forthis effect. The laser energy on target was approx-imately 390 mJ.

4. A simple model of the target charge

The charging of a laser target is a dynamic pro-cess which proceeds with several steps: (i) primaryejection of hot electrons, (ii) formation of the po-tential barrier at the target surface which confineshot electrons inside the target, (iii) gradual release

of hot electrons following the decay of the poten-tial barrier. In principle, such a process could bedescribed by kinetic simulations but in each casethis would be a major computational undertak-ing. For the purpose of quick estimates and pa-rameter optimization, it is of crucial importance tohave even a crude but computationally tractablemodel. An example of such a model appropriatefor ultrathin targets was presented in [17]. In thismodel, the central role is played by the distributionfunction f (ε, t) of the hot electrons with the en-ergy ε inside the target which is assumed to evolveaccording to:

∂tf (ε, t) =1

tlasflas (ε) θ (tlas − t)

− 1

tee (ε)f (ε, t) − gfr (ε, t) − gre (ε, t) , (1)

where tlas is the laser pulse duration, tee is the elec-tron cooling time, gfr and gre are the rates of elec-tron ejection from the front and rear side of the tar-get, and θ(t) is the step function. It is assumedthat part of the laser energy EL is converted intoa population of hot electrons, with an exponentialdistribution in energy flas(ε) ∼ exp(−ε/T0), wherethe parameter T0 is a known function of the laserintensity and wavelength. The number of hot elec-trons is N0 = ηEL/T0 where η is the laser ab-sorption coefficient. The hot electron cloud is as-sumed to expand with constant velocity v until hotelectrons reach their maximum range rmax(ε), i.e.,tee (ε) ∼= rmax(ε)/v. They are assumed to occupy acylindrical region of the target with the radius anddepth being simple functions of time and thicknessof the target. Parameters of the hot electron dis-tribution are then used to estimate the potentialbarrier Φ (t) at the surface of the target, i.e.:

Φ (t) = φth (t) + φE (t) , (2)where φth is the potential due to the negative chargeof hot electrons in the Debye layer above the tar-get surface and φE is the electrostatic potential dueto the positive charge left on the target surface bythe escaped hot electrons. The potential barrier de-termines the minimal energy of the electrons escap-ing from the target. A numerical implementation ofthis approach is realized in the Fortran code ChoCo-Lat2 [17]. The input parameters of the code arethe characteristics of the target (material, thicknessand radius), characteristics of the laser (the laserwavelength, the size of the laser spot, the pulseenergy and duration) and finally the laser absorp-tion coefficient η. The predictions of the model [17]are compared with the measurements of the targetcharge in Figs. 4 and 5 for two values of the ab-sorption coefficient η = 0.10 and 0.12. Agree-ment is acceptable, although the experiment showsa steeper dependence on the laser pulse energy.This discrepancy is understandable, however, sincein reality the absorption coefficient is an increas-ing function of the laser intensity. The choice ofthe value of the absorption coefficient requires some

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justification, since this parameter is not very wellknown, as it depends on fine details of the laser–target interaction, such as the off-peak pulse tem-poral profile and the amount of preplasma createdon the target surface. However, the same set ofparameters, including the laser absorption coeffi-cient, determines predictions for the laser acceler-ation of protons in the target normal sheath ac-celeration (TNSA) regime. In fact, the potentialbarrier that confines hot electrons inside the tar-get is the source of a quasi-static electric field thataccelerates protons and carbon ions that are al-ways present in an ultrathin — nanometer size —layer of contaminants on the surface of the target.Thus, the information on, say, the maximum pro-ton energies recorded in a given physical situationmay be used to constrain the absorption coefficientand other parameters relevant for the target chargeand hence substantially reduce the uncertainty inthe predictions for the target charge. The arrange-ment in which the target charge and the proton en-ergies are measured in the same experiment createsfavorable conditions for a stringent test of the targetcharge models.

5. Data on maximum energiesof the laser accelerated protons

The ions accelerated off the surface of the targetwere recorded in our experiment using a silicon de-tector [19] placed in a long tube protruding fromthe experimental chamber and kept under vacuum.The purpose of using such a tube was to enlargethe flight path of the accelerated ions and henceimprove the resolution of the time-of-flight spectra.The accelerated ions consist of protons and vari-ous ionization states of carbon ions but the fastestions are protons. The maximum proton velocitieswere straightforwardly estimated using the mini-mal flight time of protons, which is assumed tobe equal to the time interval between the photo-peak (tphoto), and the first ion signal (tarrival), cor-rected for the time of travel of the photopeak signal:tmin = tarrival − tphoto + L/c, where L is the dis-tance of the ion detector from the target. Figure 6shows the maximum proton energies as a function ofthe laser pulse energy. To guide the eye, the line in-dicating the best linear fit to the data is also shown:Emax = (3.6 ± 0.4)EL + (30 ± 130). The uncer-tainty in the maximum energy values is dominatedby the uncertainty in tarrival which is determinedby the oscilloscope sampling rate and the amountof noise in the ion detector signal. A conserva-tive estimate in our case would be on the orderof ±2 ns. With the typical value of the time offlight on the order of 120 ns or more, this implies atworst 4% uncertainty in the maximum proton en-ergies. The data on maximum proton energies doesshow some scatter. This scatter is a manifestationof random fluctuations in the laser–target interac-tion conditions mentioned in Sect. 3, since the phe-nomenon of laser-driven ion acceleration in this

Fig. 6. The maximum proton energies as a func-tion of the laser pulse energy, compared to predic-tions of the plasma expansion model, for two valuesof the laser absorption coefficient η = 0.10, 0.12.Also shown is the line indicating the best linear fitto the data: Emax = (3.6± 0.4)EL + (30± 130).

laser regime is well-studied and no rapidly chang-ing effects had been identified so far. Furthermore,the scatter in the maximum proton energies appearssimilar to the pattern displayed by the target chargedata which is consistent with the statement thatthe mechanisms behind these two quantities areclosely related and which supports our conclusionthat the scatter in the target charge is of randomorigin and not a manifestation of some complicatedphysical processes.

6. A model for the maximum energyof laser accelerated protons

In order to take advantage of the cross-correlationbetween the target charge deposition process andthe laser proton acceleration we need to confrontthe data on maximum proton energies with somemodel. In this paper we shall use a formulainspired by the self-similar, isothermal plasmaexpansion model [20], which was used in [21] todescribe a wide body of ion acceleration dataand more recently reviewed in [22]. Accordingto the publications, the cut-off proton energy isgiven by:

Emax = 2Tp

[ln(tp +

√t2p + 1

)]2, (3)

and the ponderomotive electron energy isdefined as:

Tp = mec2

√1 +Iλ2µm

1.37 × 1018− 1

, (4)

where me — the electron mass, c — the velocityof light, I — the laser intensity [W/cm2] and λµmis the laser wavelength [µm]. The parameter tp isthe normalized ion acceleration time:

tp =ωpitacc√2exp(1)

, (5)

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where tacc is the effective physical acceleration timeand ωpi is the ion plasma frequency for hydrogen:

ωpi =

√e2ne0mpε0

, (6)

with mp being the proton mass, ne0 =N0/ctlasSsheath — the hot electron density, andSsheath — the area of the sheath at the rear ofthe target. We have Ssheath = π (r0 + d tanϑ)

2,where r0 is the radius of the laser spot on the frontside of the target, d is the target thickness andϑ ≈ 25 is the half-divergence angle of the hotelectron stream ejected from the front side intothe target. However, differently from [21], we as-sume tacc = 3 (tlas + tmin), where tmin = 60 fs, aswas recommended in [22]. The absorption coeffi-cient η enters this formula through the expressionfor the number of hot electrons N0.

7. Comparison of model predictionswith the data

In Fig. 6, we show the predictions of this modelfor η = 0.10 and 0.12 to illustrate the sensitivityof the predictions to this parameter. A practicallylinear dependence on the laser energy is predictedin our energy range. The data seems to follow a lin-ear dependence, too, albeit with a slightly steeperslope. Similarly as in the case of the target chargemodel predictions discussed in Sect. 4, this discrep-ancy in slopes is understandable, because in realitythe absorption coefficient is an increasing functionof the laser intensity, while we used fixed values forthe whole laser energy range considered in this ex-periment. Predictions with η = 0.12 appear to lieclose to the proton data, but the predictions forthe target charge for this parameter value lie visi-bly above the data. The predictions for the targetcharge are brought close to the data by changing ηto 0.10 but then the predictions for the maximumproton energies move further below the data. Alsoin the case of dependence of the target charge onthe pulse duration shown in Fig. 5, we see that pre-dictions for η = 0.12 lie closer to the data than forη = 0.10. For both values, however, the predictionsrun below the data for the longest pulses.

In any case, obtaining an ideal fit with these twomodels does not seem possible and we did not at-tempt to further optimize η in this approach. It is,however, rather impressive that these two simplemodels predict values of proton energies and the tar-get charge which are of correct magnitude. Fur-thermore, they allow us to conclude that a sensi-ble value of the absorption coefficient for our phys-ical situation is around 0.10–0.12, which is a cleardemonstration of the advantages of measuring si-multaneously the target charge and the proton en-ergies. It remains an interesting challenge for fur-ther work to improve a simultaneous treatment ofboth these quantities in order to achieve betteragreement with the data. In particular, one could

use a more advanced model for proton accelera-tion. We used the formula of [20] because it issimple and explicit, and was found to reproducethe available data with some success [21, 22] butthis model may not be the best choice for the ratherlow laser energies explored in our experiment. Somemore advanced models for the TNSA have beenreviewed in [23, 24].

8. Measurement of electromagnetic pulses

The signal V (t) from the RB230 probe is propor-tional to the derivative of the ortho-radial compo-nent of the magnetic induction field B:

V (t) = AeqdB

dt, (7)

where Aeq is the equivalent area of the probe.In order to obtain the reading for B(t), the signalrecorded by the oscilloscope is corrected for the at-tenuation introduced by the balun (8 dB) and forthe external attenuation at the oscilloscope termi-nal, and then for the frequency-dependent attenua-tion of the coaxial cable used to connect the probeto the oscilloscope. The resulting data on dB/dt isthen numerically filtered to the service frequency

Fig. 7. (a) A typical signal for the ortho-radialcomponent of the magnetic field induction, recordedwith the RB230 B-dot probe placed at the distanceof 24 cm from the target. (b) The signal shownin (a) presented on a much shorter time scale toillustrate the fine structure of the initial spike.

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Fig. 8. The maximum value of the ortho-radialcomponent of the magnetic field induction B in-side the experimental chamber, at the distance of24 cm from the target, as a function of the laserpulse energy EL, for laser pulse duration of ap-proximately 50 fs. The line of the best linear fitis Bmax/10

−5 = (1.15± 0.09)EL + (−0.6± 0.3).

range of the RB230 probe using a 0.15–5.0 GHzband pass FFT filter and finally numerically inte-grated to yield B(t). A typical B(t) signal is dis-played in Fig. 7a. It has the form of a strong initialspike followed by decaying oscillations, lasting in allabout 100 ns.

In Fig. 7b, we show the signal presented in Fig. 7aon a much shorter time scale to illustrate the finestructure of the initial spike, the relevance of whichwas described in Sect. 2. As it can be seen, consis-tently with the shape of the return current shownin Fig. 2, the initial spike has the form of a singlefield pulsation lasting ≈ 1 ns.

Figure 8 presents the maximum value of the mea-sured ortho-radial component of the magnetic in-duction as a function of the laser pulse energy,for laser pulse duration ≈ 50 fs. The uncertaintyin these values is dominated by the uncertaintyin the probe equivalent area which is estimatedto be 4%. The data follows approximately a lin-ear dependence on the laser energy, similarly asit was observed in previous experiments of thattype [6, 14, 15]. The line of best linear fit isalso shown to guide the eye. Values on the orderof 5 × 10−5 T were measured at the top of the en-ergy range (at the distance of 24 cm from the tar-get). The corresponding values of the electric fieldstrength E may be naively estimated from the re-lation E ∼= cB to be on the order of 15 kV/m.

9. Summary and conclusions

We presented data on the target charge for 6 µmAl foil, collected at the 10 TW IPPLM laser fa-cility for the laser pulse energies in the range of130 to 400 mJ at fixed pulse duration of approx-imately 50 fs and pulse durations in the range

of 49 to 320 fs and the pulse energy on target ap-proximately 390 mJ. The data follows trends ob-served in previous experiments [3, 6, 14, 15, 17],i.e., the energy dependence is approximately linearand there is a weak dependence on the pulse du-ration. However, despite laser energies higher bya factor of 4, as compared to the previous experi-ments, the absolute values of the target charge arenot significantly higher than those previously ob-served. That is explained by a smaller laser ab-sorption coefficient in our experiment because ofa much better laser contrast at the IPPLM fa-cility and hence a much smaller amount of pre-plasma present on target at the arrival of the mainpulse. The data on the target charge was collectedin correlation with the data on maximum energiesof the laser-accelerated protons. The processes oftarget charge deposition and proton acceleration inthe TNSA regime are driven by similar physical ef-fects, namely the potential barrier at the target sur-face and are sensitive to the same parameters. A setof data on these two processes allows for a stringenttest of models of charge deposition. We illustratethis using a simple model of the target charge pre-sented in [17] and a simple formula for the maxi-mum energies of laser accelerated protons [20–22].We show that these very simple models provide pre-dictions in agreement with the obtained data, pro-vided that the laser absorption coefficient is chosenin the range η ≈ 0.10–0.12. This shows that the ob-tained data set supplies a good ground for a morerefined analysis of the target deposition mechanism.Finally, data on the maximum values of the ortho-radial component of the magnetic field inductionwere presented as a function of laser pulse energy,for the pulse duration of ≈ 50 fs. Values on the or-der of 5 × 10−5 T were measured at the top ofthe energy range.

Acknowledgments

The Polish researchers gratefully acknowledgesupport from the Polish National Science Cen-tre grant Harmonia 2014/14/M/ST7/00024. TheFrench authors acknowledge financial support fromthe French National Research Agency (ANR) inthe framework of “The Investments for the Fu-ture” Programme IdEx Bordeaux-LAPHIA (ANR-10-IDEX-03-02). V.T. acknowledges support fromthe project ELITAS (ELI Tools for Advanced Sim-ulation) CZ.02.1.01/0.0/0.0/16_013/0001793 fromthe European Regional Development Fund.

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