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High Energy Density Physics 8 (2012) 170e174

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High Energy Density Physics

journal homepage: www.elsevier .com/locate/hedp

Ion acceleration in the radiation pressure regime with ultrashort pulse lasers

Nicholas P. Dover*, Zulfikar NajmudinPlasma Physics Group, Department of Physics, Imperial College London, SW7 1AZ, UK

a r t i c l e i n f o

Article history:Received 7 February 2012Accepted 7 February 2012Available online 15 February 2012

Keywords:Radiation pressure accelerationIon accelerationHigh intensity lasers

* Corresponding author.E-mail address: nicholas.dover08@imperial.ac.uk (

1574-1818/$ e see front matter � 2012 Elsevier B.V.doi:10.1016/j.hedp.2012.02.002

a b s t r a c t

Numerical simulations of the interaction between 100 TW ultrashort (<50 fs) laser pulses and nano-metre scale carbon targets have been performed using the 2D3V PIC code OSIRIS. Different focusinggeometries (f/2 and f/0.8) were investigated, along with varying target thickness and laser polarisation, tosee the effect on the accelerated carbon ions and protons. The ions are found to be accelerated eitherdirectly by the radiation pressure of the incident radiation on the plasma, by bulk heating in the rela-tivistic transparency regime, or a combination of both. Optimum target thicknesses for maximum carbonenergies were found to be w 10 nm for the f/2 configuration and w 30 nm for the f/0.8 configuration.Despite this greater optimum target thickness, the faster focusing f/0.8 can result in a greater thandoubling in maximum ion and proton energy. Circular polarisation was found to give only a marginaladvantage in maintaining radiation pressure acceleration due to the deformation of the target duringacceleration.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

There has been much recent interest in the interaction betweenhigh intensity short pulse lasers and nanometre scale foils due tothe promise of stable acceleration of ions to high energies with highefficiency. Laser driven ion acceleration is an exciting prospect dueto the need for compact, efficient ion sources for applications suchas hadron therapy [1] and ion driven fast ignition fusion schemes[2]. A promising new avenue of research is harnessing the radiationpressure of the laser to accelerate nanometre scale overdenseplasma to high energies, similar to the light sail propulsion mech-anism proposed to accelerate spacecraft [3]. A great amount oftheoretical and numerical work has examined the scheme in one,two and three dimensions and identified the challenges to beovercome to achieve high energies [4e10]. Problems with thescheme include hydrodynamical instabilities [11] and excessiveelectron heating by the laser, especially as the target deforms due tothe transverse pulse shape [12], both of which inhibit the acceler-ation process. Initial experimental investigations of radiationpressure acceleration for thin targets have recently been reported[13] but the high-energy monoenergetic beams promised bynumerical simulation have yet to be achieved.

Radiation pressure acceleration (RPA) relies on the momentumfrom the reflected photons being efficiently coupled into an

N.P. Dover).

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overdense plasma. The radiation exerts a pressure on the plasmaelectrons, causing them to be accelerated faster than the plasmaions due to their lower mass. As the electrons move ahead of theions, an electrostatic field builds up in the region of electrondepletion, eventually balancing the radiation pressure. This quasi-static electric field then accelerates the ions, and as they moveforward the depletion region also moves forward. Hence, the laserwill bore a hole into the plasma at the hole-boring speedvb ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2IL=nimicp

[14], where IL is the laser intensity, ni the iondensity and mi the ion mass. Stationary ions ahead of this electro-static front can be accelerated by “bouncing” of the shock to twicethe hole-boring speed [15].

However, for a sufficiently thin target, the radiation pressure canact to accelerate the entire target in a coherent manner. But there isa limit to the minimum target thickness before which the targetwill not be able to transmit the full laser potential. In 1D, maximalenergies are achieved for an optimal thickness lopt ¼ lLa0(nc/ne)/p,where lL is the laser wavelength, a0 is the normalised vectorpotential, and nc/ne is the ratio of critical to initial electron density.Multidimensional considerations suggest the optimum thicknessshould be greater due to transverse motion of electrons awayfrom regions of highest intensity [6]. In the non-relativisticlimit, the estimated final speed of the ion species is given byvi ¼ ð1þ RÞsIL=ðminidcÞ, where s is the interaction time, R is thereflectivity and takes a value between 0 and 1, and d is the targetthickness. Various schemes have been suggested to reduce theeffects of target deformation and instability formation by usingshaped targets [16], shaped double layer targets [17], or multi-

Fig. 1. Charge density profiles for a) electrons, b) C6þ and c) protons at t ¼ 65 and85 fs (where peak of the laser is t ¼ 65 fs) from left to right in units of nc, fora 5 � 1020 Wcm�2 laser intensity (f/2 focusing) onto a 10 nm thick DLC foil.

N.P. Dover, Z. Najmudin / High Energy Density Physics 8 (2012) 170e174 171

species targets in which the species with the highest charge tomass ratio is protected from instabilities by the second species[18e20].

Of course for radiation pressure schemes to be effective, thedensity of the plasmamust remain above the relativistically correctcritical density, gnc [21], where the Lorentz factor is usually thatassociated with the electron quiver, which for circular polarisationwould be g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2

p.

If the density of the plasma is below this relativistic criticaldensity the electrons in the plasma are bulk heated. It has beenshown that the energy of the electrons is efficiently transferred tothe ion species when nc<ne<gnc, giving high maximum energiesbut with a characteristically exponential energy spectrum [22e24].Although using circular polarisation at normal incidence wasproposed to reduce target expansion, strong target deformationinevitably leads to density reduction and eventually laser pene-tration. For very thin targets this can become an important addi-tional mechanism for accelerating ions, though with the possiblyundesirable side-effect of increasing energy spread.

Here, we assess the possibility of using the latest generation ofhigh power (>100 TW) ultrashort pulse (<50 fs) lasers to studyradiation pressure acceleration. Such short pulse and high powersare attractive because they have the potential to offer high-energygain before instabilities and target rarefaction play a significantrole. In particular, we model the laser using the parameters of theAstra-Gemini laser system at the Rutherford Appleton Laboratory,which operates at a central wavelength lL¼ 800 nm and delivers upto w12 J on target in w45 fs full width at half maximum (FWHM),providing an average power off w250 TW.

Astra-Gemini has a typical contrast (ratio of prepulse to mainpulse intensity) smaller than 10�7. To ensure that the ultrathintarget remain intact at the peak of the laser pulse, this can beimproved by up to three orders of magnitude by employinga double plasma mirror system, with a corresponding reduction inthroughput to 48% [25]. As the final energy of the ions depends on IL(EKfI2L in the non-relativistic limit), it is desirable to use a fast-focusing parabola. With conventional f/2 focusing, the pulse can befocused to a 2.5 mm FWHM spot with 35% of the energy inside theFWHM, giving an average intensity of 5 � 1020 Wcm�2. We alsoinvestigate the potential of using faster focusing f/0.8, which wouldresult in a 1 mm spot with 35% of the energy in the FWHM, and anaverage intensity of 3 � 1021 Wcm�2.

We first discuss the results of 2D particle-in-cell numericalsimulations designed to investigate the acceleration of ions andprotons in conditions relevant to the above experimental parame-ters, with an emphasis on comparing the effect of faster focusing(section 3). Then, the effect of varying target thickness for eachfocusing optic is discussed (section 4), along with the differencebetween circular and linear polarisation in the fast-focusing regime(section 5).

2. Simulation set-up

The simulations were performed using the 2D3V particle-in-cellcode OSIRIS [26]. The grid size was 0.5 nm longitudinally and 4 nmtransversely, with 15,360 by 7040 cells, giving a simulation box sizeof 7.7 mm by 28.2 mm. The target was initialised with a top-hatdensity profile and zero temperature, which assumes perfectcontrast. The targets were modelled on diamond-like carbon (DLC),with a density of 2.7 g/cc, corresponding to an electron density,ne ¼ 466nc. DLC is considered an ideal target for RPA studies due toits high tensile strength, heat resistance and mechanical stability,which allow it to be readily made into foils of nanometre thick-nesses [13]. The ion species is initialised as fully ionised carbon 6þwith a proton contaminant in a 100:1 charge density ratio. The laser

pulse was modelled as described above, with a 2.5 mm FWHM focalspot and maximum normalised vector potential a0 ¼ 14 for a f/2focusing and a 1 mm spot and a0 ¼ 38 for f/0.8 focusing. Thepolarisation of the pulsewas circular, exceptwhen testing the effectof polarisation (see section 5).

3. Ion acceleration with fast-focusing

We first present a simulation for the f/2 parabola, with a targetthickness d ¼ 10 nm, close to the optimal thickness (loptz8 nm) forthat given intensity. The evolution of the different species in time isshown in Fig. 1. One can see many of the features commonlydescribed in simulations of radiation pressure acceleration. Thetarget expands into a bell-shape with the expansion velocity alongthe target determined by the intensity profile of the laser. Theelectron and carbon species show similar behaviour, with bothshowing the first signs of radiation pressure driven Rayleigh-Taylorlike instability modulating the beam thickness transversely. Theprotons show a slightly different behaviour, with the protons,though originating over the whole foil (which is an intrinsic prop-erty of DLC foils due to the manufacture process), eventually beingcompressed into a smaller longitudinal extent than both carbon ionsor electrons. Due to their lighter mass, the protons have actuallybeen accelerated in front of the carbon ions, and as a result exhibit

Fig. 3. Charge density profiles for a) electrons, b) C6þ and c) protons at t ¼ 42, 65 and85 fs (where peak of the laser t ¼ 65 fs) from left to right in units of nc, fora 3 � 1021 Wcm�2 intensity (f/0.8 focusing) onto a 20 nm thick DLC foil.

N.P. Dover, Z. Najmudin / High Energy Density Physics 8 (2012) 170e174172

less of the transverse modulations as compared to the carbon ions.Also due to their “squeezing” in longitudinal space, the protons allexhibit a similar accelerating field, meaning that the protons at anygiven point have very small energy spread. The final carbon andproton spectrum both integrated over all angles, and that whichwould be seen on-axis from an experimental diagnostic (usuallya Thomson spectrometer) is shown in Fig. 2. Though the total energyspread of the protons is quite large (due to the large variations inintensity across the target), by angularly selecting the protons(in particular by simulating a detector only looking at the forwardtravelling protons within an angular spread of 1 mrad) one can seethat these protons produce a naturally extremely narrow energyspread beam. The maximum energy of this beam (w 35 MeV) iscomparable to expectation for this target thickness. However it isnoticeable that there is considerable target heating, even for circularpolarisation, due to the curving of the target (so that the laser is nolonger incident normal to it all along its length). Towards the end ofthe pulse, due to both thermal and transversal expansion, the targetbecomes relativistically transparent.

Fig. 3 shows the charge density profiles of all three species atdifferent times for the f/0.8 focusing for a d¼ 20 nm target, which isclose to lopt for this focusing geometry. Although the target isinitially pushed forward by the radiation pressure of the laser, asseen on the leftmost images, before the laser intensity reaches itspeak the target breaks up, and becomes relativistically underdense.The electron density stays above nc during the peak intensity, onlybecoming underdense towards the tail end of the pulse. The elec-trons no longer comove with the ion bunch (Fig. 3a), and aretrapped in the laser field as it passes through the target, showinga pronounced bunching in phase with the laser, and gaining energyrapidly. Clearly the laser has bored through the target much morerapidly in this case, despite the thicker target, due to the extratransverse force of the tighter focused laser spot.

Despite the radiation no longer exerting a direct pressure on thetarget, the carbon species quickly expands in this relativistictransparency phase. However, due to the stationary nature of thesheath, this results in a large spatial and energy spread of thesecarbon ions. The proton species by contrast continues to be accel-erated on the edge of the carbon bunch, due to their favourablecharge to mass ratio, and so remain coherently accelerated.However, for this focusing geometry there are even larger trans-verse variations in proton energy and direction, with some of theprotons almost expelled at 90+ to the laser propagation. By the endof the interaction, the carbon species has lost all of its earlier smallenergy spread and has become broadband, whilst the energy

Fig. 2. Energy spectra from f/2 focusing onto 10 nm DLC foil showing a) The fullspectrum over all angles and b) The spectrum expected on an on-axis spectrometer.

spread of the proton emission at any given angle remains small. Theresulting carbon and proton spectrum integrated over all anglesand on-axis are shown in Fig. 4. The angular ion and proton beamprofile for a 20 nm foil is shown in Fig. 5. Although the maximum

a b

Fig. 4. Energy spectra from f/0.8 focusing onto 40 nm DLC foil showing a) The fullspectrum over all angles and b) The spectrum expected on an on-axis spectrometer.

Fig. 5. Number of particles emitted at different angles in the forward direction fora 20 nm foil in the f/0.8 geometry for protons above 5 MeV (blue (dark) dashed),protons above 30 MeV (blue (dark) solid), C6þ above 5 MeV/u (red (light) dashed) andC6þ above 30 MeV/u (red (light) solid). The proton (C6þ) beam profiles are both nor-malised to the maximum for protons (C6þ) above 5 MeV/u. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version ofthis article.)

N.P. Dover, Z. Najmudin / High Energy Density Physics 8 (2012) 170e174 173

energies are found on-axis for both carbon and protons, the particleflux is higher off axis especially for the proton beam, which showsmaximum particle flux at 45

� .

4. Optimum target thickness

To find the optimum target thickness, the target thickness wasvaried for both focusing geometries. Themaximum carbon energiesand peak proton energies as a function of target thickness areshown in Fig. 6. The maximum carbon velocity was lower than forthe protons for all but the thickest targets. This is because theprotons are pushed in front of the carbon layer and gain additionalacceleration from the expanding sheath due to hot electrons. Forthe f/2 configuration (Fig. 6a), the maximum energies were seen for5 and 10 nm DLC foils, compared to an estimated lopt z 8 nm. Forthe 5 nm target, the target becomes relativistically underdensebefore the peak of the pulse, at which point the electrons are bulkheated and transfer energy efficiently to the carbon species.Therefore, despite d<lopt, the energies achieved remain higherthan expected from a simple Coulomb explosion model. As thetargets become thicker, the ion and proton energies drop dramat-ically. The 10 nm target becomes relativistically underdense only

Fig. 6. Maximum ion energies as a function of target thickness for a) The f/2 and b) f/0.8 focusing. Circles represent peak proton energies and squares maximum carbonenergies. The dotted lines are guide lines.

after the peak of the pulse, whereas thicknesses of 20 nm and aboveremain overdense for the entire interaction. For these thicker foils(d> 20 nm), the energies therefore agreewell with those estimatedby the analytical model for RPA in the non-relativistic limit, inwhich the final energy is expected to vary as Eki f1=d2.

For f/0.8 focusing, the highest proton energies are observed ford ¼ 20 and 40 nm, although higher carbon energies are seen at40 nm rather than 20 nm. This is thus in good agreement with thethe expected loptz20 nm. This is a bonus experimentally, sincethicker targets are less fragile, and thus more likely to give repro-ducible results. Below 20 nm, the targets are too thin and becomeunderdense before the peak of the pulse, where the accelerationrelated to the relativistic transparency regime is not efficient. Thishas a particularly strong effect on the carbon ions, whilst theprotons are mostly shielded from the effect of the relativistictransparency, explaining why protons are still efficiently acceler-ated for d ¼ 20 nm whilst the C6þ are not. For the thinnest targets,the density quickly drops below nc so the ions are neither accel-erated efficiently by the radiation pressure nor the bulk heatedelectrons. For thicker targets (d>lopt), again a characteristic drop offis seen in proton and carbon energies.

In any case, one can see that the increased intensity afforded byfaster focusing more than compensates for the requirement forthicker optimum target thickness (and thus greater areal massrequired to be accelerated), This results in peak proton and carbonenergies which are a greater than factor of two improvement. Bycombining the equation for vi and lopt, it can be shown that fortargets at optimum thickness for the corresponding intensity in thenon-relativistic regime, the ion energy EkfIL. As the f/0.8 focusinggeometry simulations increased the intensity by a factor of w 7,a greater increase in maximum energy would be expected thanseen in the present simulations. This can be attributed to the fasterbreak-up of the target at d ¼ lopt for the f/0.8 geometry due to thefaster deformation of the target driven by a steeper transverseintensity gradient of the laser pulse, preventing efficient accelera-tion in the radiation pressure regime. The non-relativistic equationscaling relation works well for targets d>>lopt, where the plasmadoes not become relativistically underdense during the laserpulse and is therefore entirely accelerated by the radiation pressureof the laser.

5. Effects of polarisation with fast-focusing

At the intensities examined, it has been proposed that circularpolarisation is necessary for RPA, in order to reduce electronheating and thus keep the plasma opaque to the laser. However,despite circular polarisation the target still heats up due to vacuumheating once the target is deformed, which happens naturally dueto the radiation pressure gradient caused by the transversegaussian profile of the pulse. As circular polarisation usuallyrequires a waveplate to be inserted into the incoming beam, thiscan be expensive to implement and may degrade the quality of thebeam. Therefore, we also investigated linear polarisation for the f/0.8 focusing with a target thickness of 40 nm. Fig. 7 shows theincrease of mean electron temperature with time for both circularand linear polarisation. The circular polarisation clearly reduces theelectron heating at early times, but by the time the peak of the laserinteracts with the target (t ¼ 160(1/uL)) the electron temperaturesfor both cases are similar. Therefore, the deformation of the targethas a major impact on electron heating, especially with a tight focalspot. Comparing the on-axis simulated spectrometer from thelinear polarisation simulation (Fig. 7b) with that from circularpolarisation (Fig. 4b) shows only a slightly decreased peak energyfor the protons, and little variation on the maximum energy of thecarbon ions.

a b

Fig. 7. Results for the f/0.8 focusing on 40 nm foil, showing a) increase of meanelectron temperature with time for linear/circular polarisation, and b) final on-axis ionspectra for linear polarisation. Peak laser intensity at t ¼ 160ð1=uLÞ. Temperaturemeasurements do not include particles that have already left the box.

N.P. Dover, Z. Najmudin / High Energy Density Physics 8 (2012) 170e174174

6. Conclusion

The radiation pressure acceleration of ultrathin nanometre foilsby >100 TW ultrashort laser pulse has been examined usingnumerical 2D PIC simulations. In particular, the accelerated ionbeams from the interaction have been characterised with varyingfocusing geometries. Ions are found to be efficiently accelerated byradiation pressure acceleration at target thicknesses down to theexpected 1D optimal target thickness, lopt. The proton species isfound to be particularly coherently accelerated, since it can outrunthe carbon layer and thus ismore immune to instabilities and targetheating. Indeed the protons can be further accelerated even ford(lopt due to this bulk heating and the consequent sheathformation, which can strongly reduce the acceleration of the carbonlayer. Despite the thicker targets required to ensure that the targetdoes not become transparent too early, using faster focusingbenefits more from the increased intensity; in our simulationsshowing a greater than factor of two increase in maximum energy.Hence, this relatively simple change in focusing geometry should

be considered as away of dramatic improvement the output fromradiation pressure acceleration experiments.

Acknowledgements

This work was performed as part of the LIBRA Basic-Technologynetwork, funded by EPSRC. We acknowledge the other consortiummembers for useful discussions and suggestions. We gratefullyacknowledge the OSIRIS consortium (UCLA/IST) for the use ofOSIRIS.

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