Generation of ultraintense proton beams by multi-ps circularly polarizedlaser pulses for fast ignition-related applications

J. Badziak,1,a) G. Mishra,2 N. K. Gupta,2 and A. R. Holkundkar3

1Institute of Plasma Physics and Laser Microfusion, Euratom Association, 01-497 Warsaw, Poland2Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India3Department of Physics, Umea University, Sweden

(Received 9 December 2010; accepted 2 April 2011; published online 27 May 2011)

A scheme of generation of ultraintense proton beams relevant for proton fast ignition (PFI) which

employs multi-ps, circularly polarized laser pulse irradiating a thick (� 10 lm) H-rich target is

proposed and examined using one-dimensional particle-in cell-simulations. It is shown that a 5-ps

laser pulse of intensity� (2–5)� 1020W=cm2 irradiating the target of the areal proton

density� 2� 1020cm�2 can produce – with a high energetic efficiency – a proton beam (plasma

block) of parameters (intensity, energy fluence, pulse duration, proton energy spectrum) close to

those required for PFI. At a fixed total laser energy, the proton beam parameters can be controlled

and fitted to the PFI requirements by changing the laser intensity (energy fluence) and=or the target

thickness as well as by using a shaped (curved) target inserted into a guiding cone. VC 2011American Institute of Physics. [doi:10.1063/1.3590856]

I. INTRODUCTION

Fast ignition (FI) (Refs. 1 and 2) is an innovative

approach to inertial confinement fusion (ICF) (Ref. 3) which

differs from the conventional central-hot-spot ignition ICF

(Refs. 4 and 5) in using separate drivers for compression and

ignition of the deuterium-tritium (DT) fuel. In this approach,

the (DT) fuel precompressed by a long-pulse (ns) driver

(laser beams, X-rays) to about 1000 times the solid density

(q� 300–400 g=cm3) is ignited by a short-pulse (ps)

ultra-intense (�1020 W=cm2) particle beam (electron,1,6–8

proton,8–11 or, e.g., carbon ion12,13 beam). FI has some sig-

nificant potential advantages over conventional ICF: higher

gain, lower overall driver energy, the reduction in symmetry

requirements, and flexibility in compression drivers. The

price to be paid is the need for efficient production and cou-

pling to the compressed fuel of a particle beam of extreme

parameters.

A promising option of FI, first considered by Roth

et al.,9 relies on using a laser-driven proton beam for the fuel

ignition. Proton fast ignition (PFI) offers stiff and controlla-

ble proton transport from the source to the fuel as well as

localized and efficient proton energy deposition to the fuel.

A challenge here is achieving high laser-protons energy con-

version efficiency, reaching an extremely high beam inten-

sity and matching the proton energy spectrum to ensure an

optimal proton range in the compressed fuel.

The parameters of proton beam required for the ignition

were calculated in several papers (e.g., Refs. 10, 13, and 14

using two-dimensional (2D) radiation-hydrodynamic codes.

For a DT fuel density of� 300 to 400 g=cm3 and the proton

beam diameter of 30–40 lm they are roughly as follows:

beam energy (deposited to the fuel) of 10–20 kJ, beam inten-

sity Ip � 1020 W=cm2 (proton current density jp � a few 1013

A=cm2), proton energy fluence Fp � 109J=cm2, proton

pulse duration� 1 to 10 ps, and mean proton energy� 3 to

5 MeV for quasi-Maxwellian proton energy distribu-

tion10,13,14 or� 10 MeV for a quasi-monoenergetic energy

distribution.12,13 In particular, this means that the number of

protons heating the fuel should be above 1016, the proton

density in the beam should be about 4� 1022cm�3 and the

laser-protons energy conversion efficiency g � 15% (at igni-

tion laser energy EigL � 100 KJ). Moreover, when the proton

beam has the Maxwellian energy distribution the proton

source must be placed at a distance of less than 1 mm from

the compressed fuel to avoid power spread due to velocity

dispersion.

With the present knowledge, two laser-based methods of

proton beam generation seem to have potential to meet the

PFI requirements: Target Normal Sheath Acceleration

(TNSA) (e.g., Refs. 15–18) and skin-layer ponderomotive

acceleration (SLPA) (e.g., Refs. 17, 19–23) also referred to

as radiation pressure acceleration (RPA) (e.g., Refs. 23–30).

To produce a C ion beam for FI the laser break-out after-

burner (BOA) mechanism using an ultrathin target was also

proposed.12,31

In TNSA, protons are accelerated at the rear surface of

the foil target (usually metallic foil) by a virtual cathode

(Debye sheath) created by hot electrons produced by a laser

at the target front and penetrating through the target. As pro-

tons are accelerated mostly along the rear surface normal, by

proper curving of this surface the proton beam can be ballis-

tically focused.32,33 The proton density at the source (close

to the target) is moderate: nps � 1019cm�3 and the proton

source area, Ss, is much larger than the laser focal spot area

SL (due to lateral spread of hot electrons in the metallic tar-

get). A typical proton energy spectrum is quasi-Maxwellian

with a high energy cut off. TNSA enables generation of

a)Author to whom correspondence should be addressed. Electronic mail:

badziak@ifpilm.waw.pl.

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PHYSICS OF PLASMAS 18, 053108 (2011)

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multi-MeV proton beams of high spatial quality and with

energy conversion efficiency up to �10%.34 However, due

to relatively low proton density at the source, strong focusing

of the proton beam (ensuring an increase in the proton den-

sity by more than 1000 times) is necessary to reach the den-

sity required for the ignition. Moreover, since protons are

extracted only from a very thin (lp< 0.1 lm) layer at the rear

target surface (the areal proton density rp¼ nplp<1018cm�2),35–37 to reach the required number of protons

(� 2� 1016) the area of the proton source has to be relatively

large (Ss� a few mm2) and the distance, dsf, between

the proton source and the fuel has to be relatively long

(dsf�HSs� 1 mm). As a consequence, a proton beam of

rather narrow energy spectrum is necessary to prevent exces-

sive elongation of the proton pulse due to proton velocity

dispersion. In addition, the mean energy of protons produced

at laser intensities enabling high laser-protons conversion ef-

ficiency (�1020W=cm2) must ensure the required proton

range in the fuel. This is still an unsolved issue as energies

of TNSA protons at such intensities are too high at present to

be useful for PFI. Highly efficient generation of a proton

beam of such properties and strong focusing of the beam

without a significant energy loss seem to be the most chal-

lenging tasks for PFI with TNSA. These and other questions

related to application of TNSA-driven protons for PFI are

currently being extensively studied (e.g., Ref. 38).

In the case of application of C ions for FI, the desirable

ion energies are estimated to be� 300 to 500 MeV,12,13 so

the required number of ions NCi is a hundred times smaller

than in the case of PFI: NCi� 2� 1014. However, according

to the present knowledge, the ions of the demanded energy

spectrum can be produced only from ultrathin targets (of the

thickness LT � 0.1 lm) at laser intensities IL � 1021W=cm2

(for the laser wavelength k � 1 lm).12,31 Assuming (optimis-

tically) that all C atoms stored in the target within the laser

beam aperture are accelerated to the required energies, the

areal density of C ions at the source is limited by the value

rmaxCi � LTnc< 0.1 lm� 1023 cm�3¼ 1018cm�2. Thus, the

ion source area should be Ss> 2�10�4cm2 and the laser

power must be PL � ILSS> 200 PW. Therefore, FI with C

ions (CFI) requires a laser producing at least 100 kJ laser

energy in the pulse of duration sL< 0.5 ps and with ultrahigh

contrast ratio �1011 (to avoid a damage of the target by the

laser prepulse). At the present stage of laser technology, ful-

filling these requirements seems to be difficult. The CFI con-

cept has only started to be explored in a more detailed

manner and a lot of questions are still open here.

The indicated above difficulties in the possible applica-

tion of TNSA-driven protons or BOA-driven C ions for FI

seem to be feasible to avoid when using a high-density ion

beam driven by SLPA (RPA). In the SLPA (RPA) method,

the ponderomotive (radiation) pressure created by a short

laser pulse near the critical plasma surface drives forward

the plasma (ion) bunch of the ion density higher than the

plasma critical density, leading to the ion density of thousand

times (or more) higher than in the case of TNSA. As a result,

even at moderate ion velocities the ion beam intensities

Ii¼ niviEi and current densities ji¼ zenivi of SLPA-driven

ion beams can be very high, much higher than those of the

beams driven by TNSA. In particular, the proton beam inten-

sities required for FI seem to be attainable with SLPA with-

out the proton beam focusing or with only a slight beam

focusing. Moreover, for sufficiently long (ps) driving pulses

protons can be generated from a fairly thick (several lm or

more) solid density layer, so of high areal proton density (rp

� 1020 cm�2). It means that the proton source area can be

relatively small (Ss< 0.05 mm2) and the proton source-fuel

distance can be small as well (dsf � 0.3 mm). As a result, the

proton velocity dispersion does not cause non-acceptable

elongation of the proton pulse when using non-monoener-

getic beam. In addition, a bulk of the SLPA-produced pro-

tons has energies lower than those of TNSA protons. Thus,

obtaining the required proton range in the fuel at laser inten-

sities enabling high laser-protons conversion efficiency

seems to be easier.

However, for rather modest laser intensities (<1021

W=cm2), which are demanded to produce moderate-energy

protons required for PFI, both TNSA and SLPA contribute to

the ion acceleration process and for infrared (�1 lm) line-

arly polarized laser light, commonly employed in experi-

ments, usually TNSA dominates (especially for metallic

targets).18,36,39 In such a case the accelerated ions are spread

in space (both in the transversal and longitudinal direction)

(Refs. 16, 36, and 39) and, as a result, the ion density in the

beam and the beam intensity are much smaller than those

that could be achieved in the SLPA-dominated regime. To

increase the SLPA contribution to the acceleration process

the use of circular light polarization23–30 or a short-wave-

length laser driver40 as well as using a high-contrast laser

pulse interacting with an insulator target22,41,42 have been

proposed.

A possibility of employing the high-density neutralized

ion bunch (plasma block) driven by a ponderomotive force

for FI was suggested a few years ago (e.g. Ref. 43), but only

recently numerical simulations40,44 and some experi-

ments22,45 have provided stronger arguments supporting this

concept and allowing for its improvement.

In this paper, we demonstrate by means of particle-in

cell (PIC) simulations that using a multi-ps circularly polar-

ized laser pulse irradiating a thick (of the thickness LT � 10

lm) hydrogen target, the SLPA-driven proton beams of pa-

rameters close to those required for PFI can be produced

with a high laser-protons energy conversion efficiency. The

effect of laser intensity and the target thickness on the proton

beam characteristics and the conversion efficiency is exam-

ined in detail.

II. RESULTS AND DISCUSSION

There are a lot of works which study various aspects of

proton=ion acceleration by circularly polarized laser light

using one-dimensional (1D) or two-dimensional (2D) PIC

simulations (e.g., Refs. 23–30 and 46–50), but all these

works concern very short laser pulses (from tens to hundreds

fs) and thin (LT� 1 lm) or ultrathin (LT � 0.1 lm) targets.

Moreover, laser intensities of the driving pulses are usually

extremely high (� 1021 to 1023 W=cm2), as the main practi-

cal goal of these works is to find the best conditions for

053108-2 Badziak et al. Phys. Plasmas 18, 053108 (2011)

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production of beams of protons=ions of very high energies

(�0.1 to 1 GeV and higher), possibly with a narrow energy

spectrum. As it was mentioned earlier, PFI requires proton

beams of much lower proton energies (�10 MeV) but of

extremely high beam intensities (current densities) and beam

energy fluence. To produce such proton beams the laser

intensities should be limited to IL< 1021 W=cm2 and the tar-

get should be relatively thick (both Ip and Fp increases with

the areal proton density rp which, in turn, increases with

LT). However, to accelerate protons from a thick (�1 lm)

target with a high energetic efficiency at moderate laser

intensities, the laser pulse should be sufficiently long. A sim-

ple estimation shows that to attain Ip and Fp values required

for PFI without the proton beam focusing (or with only slight

focusing) and with the mean proton energy �Ep �10 MeV,

the thickness of a hydrogen-rich target should be LT � 10

lm and the laser pulse duration sL must be in the multi-ps

range at IL�1020 to 1021 W=cm2 (the laser energy fluence FL

� ILsL should be> 109 J=cm2). Performing 2D PIC simula-

tions of the proton acceleration process for such long driving

pulses and such thick targets, especially at realistic trans-

verse dimensions of the laser beam and the target (�100

lm), require huge computer power and very long computing

time. For this reason we restricted our simulations to the 1D

case which corresponds to the assumption of an ideally ho-

mogeneous laser beam and a planar homogeneous target.

Though such an approximation makes the examination of

some important issues like the transverse proton acceleration

stability impossible,30,50 it enables us, in particular, to esti-

mate the proton beam parameters important for PFI and their

dependence on the laser beam and the target parameters for

the above, unexplored conditions.

The simulations were performed using a fully electro-

magnetic, relativistic 1D LPICþþ code.47,51 A circularly

polarized laser pulse of 1 lm wavelength, 5 ps duration

(FWHM) and super-Gaussian of order 4 shape (Ip

! exp(t=s� 1)4 s being the pulse duration) was incident

normally on an inhomogeneous, fully ionised hydrogen

plasma layer of the density profile n(x)¼ ni(x)¼ ne(x)

defined by a linear ramp (the preplasma layer) of the thick-

ness 10k at n¼ 0 and 0 at n¼ nmax, followed by the con-

stant part of the density n¼ nmax and the thickness LT. It

was assumed that nmax¼ 40nc (nc is the critical plasma den-

sity) which corresponds roughly to the density of protons in

a CH target. The initial plasma temperature was assumed to

be 40 eV. In the computations we used 250 cells per k and

100 macroparticles of electrons and ions per cell. The total

simulation space “x” was 270k to 350k long (increasing

with the target thickness) with the 10k long vacuum region

preceding the plasma ramp. We investigated the effect of

laser intensity IL (in the range of 1020–1021 W=cm2) and

the target thickness LT (in the range of 10–80 lm) on the

proton beam parameters (peak intensity Ip, peak current

density jp, energy fluence Fp, the proton pulse shape, the

proton energy spectrum) and the laser-protons energy con-

version efficiency g at some “collecting points” (CPs)

behind the target as well as on the dynamic of changes of

the proton=electron density and the longitudinal electric

field during the acceleration process.

A. The effect of laser intensity on the proton beamparameters

Fig. 1 presents the laser-protons energy conversion effi-

ciency (g) and the proton beam energy fluence (Fp) as well

as the peak intensity (Ip) and the peak current density (jp) of

the proton beam, measured in CP placed 90 lm behind the

target, as a function of the laser pulse peak intensity IL. In

turn, Fig. 2 shows the temporal shape of the proton beam in-

tensity and current density, recorded in the above CP, for

three values of the laser peak intensity. For both the figures,

protons are produced from the target of LT¼ 45 lm. It can

be seen from Fig. 1 that the beam fluence, intensity and cur-

rent density almost linearly (though at different rates)

increase with IL up to IL¼ 9� 1020 W=cm2 and then they

slightly fall down. The efficiency starts to decrease with an

increase in IL at a few times lower intensities, however even

its minimum value is quite high, around 20%. The reason for

this decrease is too short a distance from the target to CP

because of which at IL> 2� 1020 W=cm2 all forward accel-

erated protons reach CP before a termination of the laser

pulse (t � 10ps) and they are further accelerated behind CP

(for the longer distances g attains values � 25% for

IL> 2� 1020 W=cm2). Though this distance is clearly non-

optimal, it was chosen to be comparable to the proton

source-dense fuel distance in the PFI fusion target which, for

some reasons, should be possibly short8,10,11 (also see Sec.

II C below). As seen from Fig. 2, the proton pulse recorded

in CP is very short (�1 ps), with an asymmetric shape

(of longer leading edge). It is further observed that the

proton pulse duration decreases and the peak height of the

pulse rises as the laser intensity increases. Even at the lowest

laser intensity the peak proton beam intensity is clearly

higher than 1020 W=cm2 and the peak current density is

above 1013 A=cm2.

Snapshots of the space (along x) distributions of electron

(ne) and proton (ni) densities as well as longitudinal electric

field in plasma for two values of laser intensity are presented

FIG. 1. (Color online) The laser-protons energy conversion efficiency as

well as the proton beam energy fluence, peak intensity, and peak current

density as a function of laser intensity. The proton beam parameters were

calculated for the collecting point placed 90 lm behind the target (lCP¼ 90

lm) and for the target thickness LT¼ 45 lm.

053108-3 Generation of ultraintense proton beams by multi-ps circularly polarized laser pulses Phys. Plasmas 18, 053108 (2011)

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in Fig. 3. For both these intensities almost all the electrons

and protons stored in the target (of 45 lm thick) are acceler-

ated forward by the piston of the electric field produced at

the skin layer by the laser-induced ponderomotive force.

Excluding the skin-layer region, where the differences in the

electron and proton densities are distinct, in the remaining

part the proton density distribution almost ideally reproduces

the electron density distribution (the electric field inside the

plasma is small). Thus, the ponderomotive force accelerates

actually a highly neutralized dense plasma bunch (plasma

block20,43). It is an important feature of the accelerated pro-

tons which can benefit in easier focusing and transporting

the proton beam towards the DT fuel core (see further). Fig.

3 also proves that, in the considered case, SLPA clearly dom-

inates over TNSA as the longitudinal electric field at the

leading edge of the plasma bunch, which is responsible for

the TNSA mechanism of ion acceleration, is negligibly small

in comparison with the field in the skin-layer region. As a

result, the number of ions pulled by this “TNSA field” is also

much smaller than the number of ions pushed by the ponder-

omotive piston. This feature can be seen also in Fig. 7 for the

targets of other thicknesses. However, there are a few groups

of protons inside the bunch which are moving with different

velocities. It can be seen even better in Fig. 4, where the pro-

ton energy spectrum recorded at CP is shown for two instants

corresponding to the proton pulse leading edge and (roughly)

to the proton pulse peak. We observe a multi-peak structure

of the spectrum at the earlier time and a tendency to trans-

form the spectrum in a two-peak structure (the spectrum

shape stabilizes only when the laser pulse terminates). This

double peak structure indicates the existence of multi–stage

acceleration.24,47 The process of multi-stage acceleration

can be visualized as follows. During the initial process of

ion acceleration the maximum ion velocity continuously

increases with time up to some time and then remains almost

constant for certain period (e.g., see Fig. 5 of Ref. 47). In

this interval the number of ions acquiring this maximum ve-

locity increases and saturates leading to the first peak in the

ion energy spectrum. This can be considered as completion

of a first stage. After this time, the maximum velocity

sharply increases while the number of ions having this higher

velocity decreases indicating that a fraction of first stage

accelerated ions gets further acceleration. Subsequently this

number slowly builds up with time and saturates leading to

the second peak. As only a fraction of the first stage acceler-

ated ions get second stage acceleration we observe a double

peak structure in ion energy spectrum. Such a two-peak pro-

ton spectrum (corresponding to the asymmetric proton pulse

shape as in Fig. 2 or to a double proton pulse) can be

FIG. 2. (Color online) The temporal shape of intensity (It) and current den-

sity (jt) of the proton beam recorded at lCP¼ 90 lm for various laser inten-

sities. LT¼ 45 lm.

FIG. 3. (Color online) Snapshots of the proton density (ni), electron density

(ne), and longitudinal electric field (Ex) distributions for proton beams driven

by a laser pulse of moderate (a and b) and high (c and d) laser intensities.

LT¼ 45 lm, s � 3.3 fs is the laser wavelength period.

053108-4 Badziak et al. Phys. Plasmas 18, 053108 (2011)

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beneficial in application of the proton beam for PFI, as it

allows for better concentration of energy deposited by the

proton beam to the fuel (the faster protons increase the fuel

temperature which results in an increase in the range of the

slower protons in the fuel).10,13

B. The effect of the target thickness on the protonbeam parameters

The areal proton density of the proton source increases

with an increase in the proton target thickness LT which, in

general, should result in a growth of the total number of

accelerated protons. However, other proton beam parameters

and the laser-protons energy conversion efficiency depend

on LT in a non-monotonic way, therefore to maximize these

parameters the target thickness should be well matched to

the laser beam parameters. The dependence of the conver-

sion efficiency as well as the proton beam energy fluence,

peak intensity, and peak current density on the target thick-

ness in CP situated 90 lm behind the target and for

IL¼ 2� 1020 W=cm2 is presented in Fig. 5. For the assumed

value of laser energy fluence of FL � 1GJ=cm2, the number

of protons stored in the thinnest target (of LT¼ 10 lm)

seems to be to small in comparison with the number of accel-

erating photons (not all the photons could be efficiently used

for the proton acceleration) and, as a result, the proton beam

fluence and the conversion efficiency are clearly lower than

for the thicker targets. In the relatively broad LT range

of� 20 to 50 lm, Fp and g slightly depend on LT and only at

LT¼ 80 lm (not shown here) the values of these parameters

fall to the values about twice as small (the number of protons

in the target is too high). The dependences of the proton

beam peak intensity and current density on LT are a bit stron-

ger and some optimum target thicknesses can be distin-

guished. It is worth noting that in the broad range of LT the

conversion efficiency is >20% and Ip and jp are well above

1020 W=cm2 and 1013A=cm2, respectively. The proton pulse

duration measured in CP significantly changes with the tar-

get thickness and for the 45-lm target, it is three times lon-

ger than that for the 10-lm target (Fig. 6).

Snapshots of the ni(x), ne(x), and Ex(x) distributions for

two values of LT are shown in Fig. 7. In spite of the signifi-

cant difference in the thicknesses, both targets are acceler-

ated in the form of near-solid density plasma block by the

piston produced at the skin layer by the ponderomotive

force. However, at the assumed value of FL�1 GJ=cm2, the

average velocity (energy) of protons in the plasma block

FIG. 4. (Color online) The proton energy spectra recorded at lCP¼ 90 lm

for two different laser intensities. LT¼ 45 lm.

FIG. 5. (Color online) The laser-protons energy conversion efficiency as

well as the proton beam energy fluence, peak intensity, and peak current

density as a function of the target thickness. lCP¼ 90 lm, IL¼ 2�1020

W=cm2.

FIG. 6. (Color online) The temporal shape of intensity (It) and current den-

sity (jt) of the proton beam recorded at lCP¼ 90 lm for various target thick-

nesses. IL¼ 2�1020 W=cm2.

053108-5 Generation of ultraintense proton beams by multi-ps circularly polarized laser pulses Phys. Plasmas 18, 053108 (2011)

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formed from the 80 lm thick target is too small to penetrate

effectively the dense DT fuel core, so we need either a thin-

ner target or higher laser fluence to produce the required pro-

ton energy spectrum.

The proton energy spectra in CP for the target thickness

of 10 lm and 30 lm are presented in Fig. 8. For the thicker

target, the maximum and the mean proton energy are about

twice as small as those for the 10-lm target. Thus, the

changing of the target thickness can be one of the ways of fit-

ting the proton energy spectrum to the required one.

C. Relevance of the proton beam parameters to thePFI requirements

Let us try to estimate whether parameters of the proton

beam produced from a thick H-rich target by multi-ps laser

pulse could meet the PFI requirements. We assume that the

energy of a 5-ps, circularly polarized laser pulse of k � 1 lm

is equal to 100 kJ, which is usually assumed as a reference

value for the short-pulse FI driver.7,8,10,11 The laser pulse

interacts with a curved H-rich target of the areal proton den-

sity rp¼ 2� 1020 cm�2 (which corresponds to the CH target

thickness LT � 40lm or the CH2 target thickness LT �25lm). The target is inserted into a guiding cone with the

cone tip placed at the direct vicinity of the dense DT fuel

core7,8,11 (Fig. 9). The (internal) diameter of the cone tip is

dt¼ 40 lm and the distance from the target to the cone tip is

lTt � 100 – 150 lm. The average target diameter is

dT¼ 100lm, so the total number of protons stored in the tar-

get is � 1.6� 1016. The diameter of (possibly homogeneous)

laser beam is assumed to be dL� dT (Fig. 9), therefore the

(average) laser intensity on the target is IL � 2.6� 1020

W=cm2.

According to the plots presented in Figs. 1 and 5, at the

distance from the target equal to lTt the laser and target pa-

rameters assumed above should result (in 1D geometry) in

the proton beam parameters as follows: the energy fluence Fp

� 0.3 GJ=cm2, the peak intensity Ip � 4� 1020 W=cm2, the

peak current density jp � 5� 1013 A=cm2, the pulse duration

FIG. 8. (Color online) The energy spectra of protons produced from the tar-

gets of different thicknesses. lCP¼ 90 lm, IL¼ 2�1020 W=cm2.

FIG. 7. (Color online) Snapshots of the proton density (ni), electron density

(ne), and longitudinal electric field (Ex) distributions for proton beams pro-

duced from the targets of different thicknesses. IL¼ 2�1020 W=cm2.

053108-6 Badziak et al. Phys. Plasmas 18, 053108 (2011)

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si � 1 ps and the total proton energy Etot � 20 kJ. However,

in the proposed scheme (Fig. 9) the target front surface is

curved (say, it is quasispherical), so the surface of the pon-

deromotive piston driving protons (plasma block) is curved

as well (though, in general, with the curvature radius differ-

ent from that of the target surface).52,53 As a result, at a

proper shape of the target front surface, the proton beam can

be convergent,52,53 with a focal spot placed, e.g., in the DT

fuel just behind the cone tip. However, due to possible inho-

mogeneity of the laser beam and 2D=3D phenomena, part of

protons can be accelerated along trajectories different from

those of the main proton flux and without the guiding cone

they could be lost. Basically, the cone can help to minimize

these proton losses. It can be supposed taking into account

the fact that the ponderomotive piston do not accelerate sin-

gle charged particles but actually, a highly neutralized

plasma bunch. Since the hydrodynamic expansion time for

this plasma is much longer than the (ponderomotive) acceler-

ation time tacc� 10ps (the plasma temperature is in the

multi-keV range for a circularly polarized laser driver),36

such plasma can be effectively transported in a guiding chan-

nel, as it was demonstrated recently.54 Anyway, we assume

that 20% energy of the proton beam is lost due to imperfect

proton beam focusing and guiding inside the cone. As a

result, the proton beam parameters at the cone tip of dt¼ 40

lm (at the entrance to the dense DT fuel core) are roughly as

follows: Fp � 1.5GJ=cm2, Ip � 2� 1021 W=cm2, jp �2.5� 1014 A=cm2, and Etot � 16 kJ. Moreover, the proton

energy spectrum has a two-peak structure with the first peak

of mean energy �10 MeV and the second one of mean

energy �5 MeV (see Figs. 4(a) and 8(b)). We see that all

these parameters are fairly well matched to the typical PFI

requirements (see Introduction and Refs. 8, 10–14), in spite

of the fact that we used rather arbitrary (non-optimized) and

rough numbers as the laser and target parameters and only

slight focusing of the proton beam was assumed. The

assumed laser parameters are not very demanding (the focal

spot diameter �100 lm is relatively large, the laser

power� 20 PW is moderate and the laser contrast ratio� 108

to 109 seems to be acceptable) and are within the reach of

emerging laser technologies. The proton target construction

(including the guiding cone) is relatively simple and work-

able with current technology (at least with non-cryogenic

proton target materials like, e.g., CH2 or UH3). Due to its

large thickness, the target is expected to be resistant to a

damage or non-acceptable degradation by a (moderate) laser

prepulse or X-rays (a moderate heating of the target can

result even in an improvement of its performance as opposed

to, e.g., TNSA targets). Obviously, like in other FI scenarios

with a guiding cone, a survival of the cone during the fuel

implosion is an open question.

As was shown in Secs. II A and II B, the proton beam

parameters can be controlled by changing laser and target pa-

rameters, in particular, the laser intensity, the laser energy

fluence, and the target thickness (at fixed energy of the laser

driver, the first two can be changed – within some limited

range – by variation of the laser pulse duration and=or the

focal spot diameter). We could see that the dependence of

the proton beam parameters on the above laser and target pa-

rameters is sufficiently strong to ensure effective control of

the proton parameters in this way. On the other hand, the

sensitivity of the proton parameters to small changes in the

laser=target parameters is low. Thus, the scheme using a

multi-ps circularly polarized laser driver and a thick H-rich

target seems to be sufficiently flexible to fit the proton beam

parameters to the PFI requirements and, simultaneously,

quite stable in regard to small changes of its basic parameters

(at least in 1D geometry). As such, it could be considered as

(we believe) a well established starting point for more thor-

ough and systematic, mostly 2D studies of the scheme with

possibly realistic target=laser parameters and geometries.

There are several key issues that need to be investigated: (1)

a transverse (2D=3D) stability of the proton (plasma) accel-

eration driven by an inhomogeneous laser beam, (2) focusing

the plasma block driven by a ponderomotive piston, (3) guid-

ing the accelerated plasma block in the cone, and (4) proton

acceleration in a two-ion-species H-rich target (like CH2,

UH3, solid H2O, etc.) with a possible inclusion of proton

energy losses in the target due to collisions. Unfortunately,

due to long laser pulse duration and large transverse dimen-

sions of the considered scheme, numerical investigations of

these issues (at least (1), (2), and (3)) require a very high

computing power and are possible only with the use of big

computers.

It should be mentioned that the transverse stability of

proton acceleration using thin or ultrathin targets and ultra-

short (�100 fs), circularly polarized laser drivers has been

extensively studied recently (e.g., Refs. 30 and 50). It was

found out that in such conditions Rayleigh-Taylor-like (RT)

instabilities can significantly influence the acceleration,

although there exist regimes in which the instabilities can be

avoided.30,50 This issue is of a key importance for potential

applications of the accelerated protons in cancer therapy or

accelerator technology, where the proton beam has to cover

a long distance (� than the proton source size) and can be

completely destroyed by the instabilities before reaching an

FIG. 9. (Color online) An idea of PFI with the proton beam (plasma block)

produced by a multi-ps laser pulse from a thick, curved target placed inside

a guiding cone (see the text for details).

053108-7 Generation of ultraintense proton beams by multi-ps circularly polarized laser pulses Phys. Plasmas 18, 053108 (2011)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp

object. However, in the case of our scheme, both a physical

situation and the practical meaning of possible transverse

instabilities in the acceleration process are different than in

the above case. First, the proton beam (plasma block) propa-

gation distance, the target transverse size, and the target

thickness are comparable to one another (they differ within a

factor 2–4), so a significant distortion of the proton beam by

the instability is much less probable than for the ultrathin tar-

get. Secondly, even if the proton beam is highly inhomoge-

neous it can still efficiently heat the DT fuel,8 provided that

the energy loss while transporting such inhomogeneous

beam inside the cone is not too large. In spite of the fact that

consequences of the possible RT instabilities and other

2D(3D) effects in our scheme are not expected to be so dra-

matic as in the case of ultrathin targets and=or long propaga-

tion distances, these effects can potentially lead to a growth

of energy loss (due to an increase in the proton beam angular

beam divergence) and to a decrease in the laser-protons

energy conversion efficiency (due to the non-linear depend-

ence of the efficiency on laser intensity). Though we believe

that taking into account these effects in more thorough nu-

merical studies will not result in changing general conclu-

sions formulated in our work, this issue needs a detailed

quantitative analysis and conditions for minimizing a possi-

ble energy loss should be determined.

Another important issue that needs a comment is the tar-

get composition. Our simulations were performed for the

hydrogen plasma, so the obtained (quantitative) results can

be considered as related to the case of using pure hydrogen

solid-state (criogenic) target. Though manufacturing such

targets and their coupling to DT criogenic fusion targets with

a cone seem to be feasible with present technology, much

more simpler would be using the targets made of hydrocar-

bons (e.g., CH2 ) or hydrides (e.g., CaH2, ErH3, or UH3).55

However, in a two-ion-species target a part of laser energy is

transferred to heavier ions and, as a result, the efficiency of

proton acceleration is smaller than for a pure hydrogen. To

minimize this decrease in the proton acceleration efficiency,

the target material needs to be properly selected. It was

shown55,56 that a division of the laser energy between accel-

erated protons and heavier ions significantly depends on the

z=A ratio (z is the charge state and A is the mass number of

the ion, respectively) and the target mass density, and for the

heaviest hydrides (UH3, ErH3) even � 90% of the total trans-

ferred energy can be utilized for acceleration of protons.55

Very briefly, it is due to the fact that at very high A (very

low z=A ratio) and moderate laser intensity the moving elec-

trostatic potential of the ponderomotive piston is too low to

reflect heavy ions and it penetrates through the target leaving

these ions immovable. As a result, the piston energy is used

to accelerate only protons. Thus, by a careful optimisation of

the target composition, reaching the proton beam parameters

close to those achieved with the pure hydrogen target seems

to be possible.

III. CONCLUSIONS

The use of multi-ps, circularly polarized laser pulse and

a thick (�10 lm) H-rich target for production of proton

beams relevant for fast ignition and related applications

was proposed and examined using 1D PIC simulations.

It was shown that a 5-ps laser pulse of intensity� (2 – 5)

� 1020 W=cm2 irradiating the target of the areal proton

density� 2�1020 cm�2 can produce – with a high energetic

efficiency �20% – the proton beam (plasma block) of param-

eters (intensity, current density, energy fluence, pulse duration,

proton energy spectrum) close to those required for PFI. At a

fixed total laser energy, the proton beam parameters can be

controlled and fitted to the PFI requirements by changing

the laser intensity (energy fluence) and=or the target thickness

as well as by using a shaped (curved) target inserted into

a guiding cone. The results of our simulations seem to consti-

tute a well established starting point for more thorough and

detailed studies of the considered scheme addressing such

issues as 2D=3D effects, proton beam focusing and guiding

inside a cone or employing targets with more realistic target

composition.

ACKNOWLEDGMENTS

This work was supported in part by the HiPER project

under Grant Agreement No 211737 as well as by the Minis-

try of Science and Higher Education (MNiSZW) Poland

under Grant No. N202 207 438.

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053108-9 Generation of ultraintense proton beams by multi-ps circularly polarized laser pulses Phys. Plasmas 18, 053108 (2011)

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