of the light ion beam, since it can acquire an excess of electrons leaked from ... that a monoenergetic proton beam with energy of 18MeV is produced by ... When the foil is of nm-scale thickness, less than (or close to) its skin depth ls, the laser .... larger charge-mass ratio tend to move faster and the heavy C6+ slower, so the ...
Radiation Pressure Acceleration of Ion Beams from Nanofoil Targets: The Leaky Light-Sail Regime B. Qiao,1, 2 M. Zepf,2 M. Borghesi,2 B. Dromey,2 A. Karmakar,1 M. Geissler,2 and P. Gibbon1 1
J¨ ulich Supercomputing Center, Forschungzentrum J¨ ulich GmbH, D-52425, J¨ ulich, Germany 2
Department of Physics and Astronomy,
Queen’s University Belfast, Belfast BT7 1NN, UK
Abstract A new ion radiation pressure acceleration regime, the leaky ”Light-sail”, using sub-skin-depth nanometer foils irradiated with circularly polarized laser pulses is proposed. In the regime, the foil is partially transparent, thus continuously leaking electrons out by the transmitted laser field. With this feature, a multi-species nanofoil provides a configuration to achieve a stable acceleration of the light ion beam, since it can acquire an excess of electrons leaked from those associated with the heavy ions as a supplement to avoid Coulomb explosion. It is shown by 2D PIC simulations that a monoenergetic proton beam with energy of 18MeV is produced by circularly polarized lasers at intensities of just 1019 W/cm2 . 100MeV proton beams are obtained by increasing the intensities to 2 × 1020 W/cm2 . PACS numbers: 52.38.Kd, 52.50.Jm, 29.25.-t, 52.65.Rr
1
Multi-MeV ion acceleration from laser-irradiated solid foils has become a highly active field of research over the past few years [1–6]. The wide applications [7] include tumor therapy, radiography, laser-driven fusion, and so on. Most of the applications require a high-energy ion beam with large particle number and monoenergetic spectrum. Radiation pressure acceleration (RPA)[3–6] using circularly polarized (CP) laser pulses has emerged as a promising route to obtaining such high-quality ion beams in a much more efficient manner, compared to the target normal sheath acceleration (TNSA) [1, 2]. The simplest version of RPA scheme is that a layer of ions and a compressed electron layer are synchronously boosted by the laser pulse as if they constituted a quasineutral plasma slab, where equilibrium between the electrostatic and radiation pressures holds. The final peak energy of ions is determined by the laser pulse energy and the slab total mass. By decreasing the foil thickness, enhancements in both the peak energy and conversion efficiency can be achieved due to the target’s limited mass. Recently, the acceleration of high-energy ion beams from nanometer (nm) foils has aroused particular interest [8, 9]. Depending on the foil thickness l0 , three regimes can be identified for the RPA scheme. First, if the foil is semi-infinite, ions experience a ”hole-boring” (HB) phase of RPA [10, 11]. The early states of electron and ion density profiles and the electrostatic field Ez before ions start to move can be described by Fig. 1(a). Electrons are piled up into a compressed layer of thickness d2 ∼ ls , where ls = c/ωpe is the foil skin depth of electron density ne0 and ωpe = q
4πne0 e2 /me . Because this HB is continuous, all ions are accelerated by Ez in a pistonlike
manner by repeated reflections, finally reaching a maximum velocity vi,max = 2vb , where √ vb /c = a/(2 mi ni ); mi , ni and a are the ion mass, density and normalized laser amplitude respectively. To obtain high-energy ion beams a very high-intensity laser is required as vi,max does not increase with time after the HB has saturated. Second, if the foil is of micron-scale (hundreds of nm to microns) thickness and l0 > ls , electrons pile up at its rear side in a thin layer of d2 l0 , and are held inside the foil because the laser field vanishes there –Fig. 1(b). The HB finishes prematurely as laser pulse quickly punches through the foil. Only the ions at the foil rear side initially located in the compression region (d1 < z < l0 ) are bunched into a high-density layer by the leading edge of Ez , while the others undergo Coulomb explosion by the trailing part of Ez [12]. Afterwards the electron and ion layers combine together as a quasi-neutral plasma slab pushed by laser pulses, known as ”Light-sail” (LS) RPA regime [13]. However, in the real multi-dimensional 2
case, this plasma slab loses electrons in the latter time due to transverse instabilities [4, 6, 14] at the laser-facing surface, which quickly becomes positively charged. As a result, the ion beam debunches by a rapid Coulomb explosion and cannot be accelerated to high-energy. A stable acceleration can only be achieved when the laser intensity I0 ≥ 1022 W/cm2 [6]. When the foil is of nm-scale thickness, less than (or close to) its skin depth ls , the laser field does not significantly decay within the foil and so all electrons will be accelerated by the transmitted field. Electrons near the rear side will be leaked into vacuum since the laser field does not vanish there. The equilibrium states depicted in Fig. 1(a) or (b) break down. And therefore, the previous theoretical models of either ”phase-stable” [5, 9] or ”cyclic” [4, 12] accelerations no longer work for this new regime, which we call ”leaky light-sail (LS)” RPA. In this Letter, we present for the first time theoretical and numerical studies on this new Leaky LS regime, where a sub-skin-depth nm foil (nanofoil) is irradiated by CP laser pulses. A theoretical model is described, consistent with two-dimensional (2D) particle-incell (PIC) simulations. Exploiting the feature that the foil (slab) constantly leaks electrons from its rear, a multi-species nanofoil is suggested in which the light ions can experience a stable acceleration in the real multi-dimensional geometry by low-intensity CP laser pulses. The idea is verified by 2D PIC simulations with a hydrocarbon (CH) foil. A monoenergetic proton beam of energy 18MeV is produced by CP lasers at intensity I0 of 1019 W/cm2 . By increasing I0 to 2 × 1020 W/cm2 , 100MeV proton beam are obtained by a stable acceleration. At first we run 2D PIC simulations for a single-species nanofoil to get an insight into this new regime using the code ”ILLUMINATION” [15]. We choose a CP laser pulse with wavelength λ = 1.0µm and intensity I0 = 3.4 × 1019 W/cm2 (a0 = 5). The pulse is incident along z axis from z = 0 with a flattop envelope of 1-cycle rise time and 40-cycle plateau. A fully ionized hydrogen foil with electron and proton densities ne0 = nip0 = 200nc locates at z0 = 3µm. The foil thickness l0 = 8nm is less than ls = 11.3nm but thick enough to satisfy (1/π)(nc /ne0 )a0 λ ≤ l0 < ls
(1)
ensuring not all electrons are blown-out [6, 12]. The simulation space (15×12µm) is composed of 30000×3000 cells along z and x directions. Each foil cell is filled with 800 quasiparticles. Figure 2 (a)-(d) shows density maps of electrons ne and protons nip at time t = 12.5, 20, 34 and 80fs. Before ions start to move – Fig. 2(a) – we see that all foil electrons are accelerated by the transmitted laser field: electrons at the front side are pushed inward, leaving behind 3
an electron depletion region; however, electrons in the rear side are dragged into vacuum as the laser field does not vanish there, forming an electron leakage region; electrons are much less compressed. The longitudinal profiles of ne , nip and Ez in Fig. 2(e) clearly correspond to the schematic Fig. 1(c) rather than 1(a) or 1(b). Three charge seperation regions can be divided: the (i) depletion (0 < ξ < d1 ), (ii) quasi-neutral (d1 < ξ < l0 ) and (iii) leakage (l0 < ξ < l0 +d3 ) regions, where ξ = z −z0 . The electrostatic field Ez can be also modeled as
Ez =
E0 ξ/d1
if 0 < ξ < d1 if d1 ≤ ξ ≤ l0
E0
E
0 [1−(ξ −l0 )/d3 ]+Ebo (ξ −l0 )/d3
(2)
if l0 < ξ ≤ l0 +d3
including the leading and trailing edges and the uniform part, where E0 = 4πene0 d1 is the maximum value of Ez . Ebo 6= 0 is the contribution induced by the electrons already blownout of the three regions with ξ > l0 +d3 , so Ez at ξ = l0 +d3 is still not zero. Note that the exact profile of Ez obtained here cannot be seen in the low-resolution simulations [9]. Figure 2(b)-(d) shows acceleration of the latter time. We see that only the protons located in the quasi-neutral region are bunched into a thin layer by the uniform part of Ez (Ez = E0 ), while those in the depletion region are debunched due to Coulomb explosion by the trailing edge of Ez . Meanwhile, only the electrons in the quasi-neutral region are accelerated as a bunched layer while those in the leakage region are blown-out rapidly because no ions exist there. These ion and electron layers combine together to form a plasma slab of even less thickness d2 = l0 −d1 < l0 < ls , shown in Fig. 2 (b). Therefore, in contrast to the purely LS regime, this slab not only loses electrons at the laser-facing surface due to transverse instabilities but also continuously leaks electrons from its rear by the non-vanished laser field [also 2(b)], which changes quickly from being negatively [2(f)] to positively [2(g) and (h)] charged. As soon as the slab is positively charged, the local Ez acquires a purely trailing edge, ultimately resulting in the broad energy spectrum in Fig. 4(c) by the dashed line. The above arguments imply that ion RPA in the Leaky LS regime is even worse than the purely LS regime if a single-species target is used. However, this negative result gives us a key physical insight of this new regime which may be turned to our advantage. The feature that the plasma slab leaks electrons from its rear during the acceleration may be useful for RPA of protons in a multi-species nanofoil target with mixture of heavy and light ions (protons). It is obvious that a heavy ion with higher charge state Z is accompanied with Z times more electrons than a proton. If a multi-species nanofoil is used, the electrons 4
leaked from those accompanied with the heavy ions may be just acquired by the proton layer as a large supplement because the latter only needs 1/Z electrons for charge balance. As a result, despite the loss of electrons due to leakage and/or transverse instabilities, the proton layer may remain surrounded by an excess of electrons to keep locally negatively-charged space, which can avoid Coulomb explosion and preserve stable RPA. Therefore, we expect protons from a multi-species nanofoil in the Leaky LS regime to experience a stable RPA in the real multi-dimensional geometry, so that a high-energy monoenergetic proton beam can be obtained by much lower-intensity lasers, in comparison with the purely LS regime. To check this idea by 2D simulations, we take a hydrocarbon (CH) nanofoil and assume it to be fully ionized into C6+ and H+ . And to keep ls identical to that of the single-species case, we choose the same electron density ne0 = 200nc and thinkness l0 = 8nm. The densities of H+ (proton) and C6+ are taken respectively as nip0 = 4.1nc and nic0 = 32.65nc with their ratio nip0 : nic0 ≈ 1 : 8. All the laser and other parameters are also the same. Figure 3 (a)-(h) also gives density maps and longitudinal profiles of electrons ne , protons nip and C6+ ions nic at t = 12.5, 20, 34 and 80fs. The early state of ne and Ez in (a) and (e) are very similar to that in the single-species case, which is also composed of the above three typical regions. The multi-species effect does not influence the electron dynamics. Similarly, we see only the protons and C6+ ions initially in the quasi-neutral region are bunched into thin layers by the quasi-uniform part of Ez [Fig. 3(e)]. Protons with larger charge-mass ratio tend to move faster and the heavy C6+ slower, so the proton layer moves ahead of the C6+ layer. If the accompanied electron layer is stably pushed without debunching or leakage, where equilibrium between the electrostatic and radiation pressures holds, as in the HB regime [11], we know that the faster proton (slower C6+ ) layer will be pulled back (forward) by the space-charge field redistribution and finally the entire plasma slab (the proton, C6+ and electron layers) will move together at the same velocity 2vb , which only depends on the total mass of the slab, irrespective of their individual charge and mass. However, here in the Leaky LS regime the laser field does not significantly decay within the slab, which cannot be balanced by the electrostatic field. The electron layer debunches and constantly leaks electrons [Fig. 3(b) and (f)]. This debunching of the electrons leads to an almost complete separation of the C 6+ and proton layers, also see 3(b) and 3(f). Moreover, due to the loss of electrons, the C6+ layer has insufficient electrons for charge balance and the local space charge becomes positive [Fig. 3(g)]. By contrast, the proton layer 5
is surrounded by an excess number of electrons keeping locally negative charge space [also 3(g)] due to the supplement of electrons leaked from those associated with the C6+ layer. In other words, the proton layer ”borrows” electrons from the C6+ layer. Therefore, the proton layer maintains stable RPA by the leading edge of Ez while the C6+ layer undergoes rapid Coulomb explosion by the trailing part of Ez [see Fig. 3(d) and (h)]. Figure 4 (a) shows the high-energy proton beam at t = 130fs when the laser pulse is over. We see that a quasi-monoenergtic proton beam is produced with density about 0.25nc and peak velocity vz ' 0.18c. The energy spectra of protons and C 6+ are shown in 4(b). The proton beam has a peak energy of 18MeV with FWHM of 5MeV. The particle number in the beam within |x| < 1µm is about 108 and the estimated conversion efficiency is about 10%. The evolutions of both the peak Ez acting on and the peak energy of the proton beam are given in 4(c) and 4(d) for the whole acceleration process. We can see the typical features of RPA scheme: Ez decreases very slowly at a later time as it is in balance with the radiation pressure; and the peak proton energy at a later time scales as t1/3 , in consistent with the typical analytical scaling of RPA [4, 6]. This proves that the proton beam does experience stable RPA other than direct Coulomb explosion (DCE) [16]. The DCE scheme [16] is proposed for the case of linearly polarized lasers, where the laser intensity should be extremely high to the order of 1022 W/cm2 so that almost all the electrons are blown-out and the ion core is accelerated by Coulomb explosion, in contrary to condition (1) here. Note that the proton acceleration time here is limited by the laser pulse duration. For a laser pulse with finite duration τL , the upper limit of proton energy can be estimated as 2η 2 EL2 /(2ηEL + Ni mi c2 )Ni , where EL = I0 SτL is the laser pulse energy, S are its transverse area; Ni and η are the particle number and conversion efficiency. So one can obtain higherenergy proton beams, such that 100s of MeV, by using a multi-species nanofoil irradiated with CP lasers of longer pulse durations and intensities experimentally accessible with existing laser systems. For example, we choose a CP laser pulse with flat-top of 70-cycle plateau and intensity 2 × 1020 W/cm2 to irradiate on a CH nanofoil of l0 = 6.4nm and ne0 = 600nc , where the condition (1) is satisfied. The density ratio of H+ to C6+ is 1 : 12. Figure 4 (e) and (f) shows at t = 240fs a monoenergetic proton beam of energy about 100MeV and densities 0.15nc is obtained. The particle number in the beam is about 107 . To summarize, a new leaky light-sail regime of ion RPA from sub-skin-depth nanofoil targets by CP laser pulses has been proposed. Using the feature of this regime that the 6
foil and plasma slab constantly leak electrons from its rear, a multi-species nanofoil is suggested to achieve a stable RPA of the light ions in the real multidimensional geometry by low-intensity lasers. Two-dimensional PIC simulations show that a monoenergetic proton beam of energy 18MeV is produced by irradiation of a 8nm CH nanofoil with CP lasers at intensity 3.4×1019 W/cm2 . 100MeV proton beams are obtained by CP lasers at intensities of 1020 W/cm2 , which are experimentally available with existing laser system.
[1] E. L. Clark et al., Phys. Rev. Lett. 84, 670 (2000); A. Maksimchuk et al., ibid. 84, 4108 (2000); R. A. Snavely et al., ibid. 85, 2945 (2000). [2] B. M. Hegelich et al., Nature 439, 441 (2006); H. Schwoerer et al, Nature 439, 445 (2006). [3] A. Macchi et al., Phys. Rev. Lett. 94, 165003 (2005). [4] A. P. L. Robinson et al., New. J. Phys. 10, 013021 (2008). [5] X. Q. Yan et al., Phys. Rev. Lett. 100, 135003 (2008). [6] B. Qiao et al., Phys. Rev. Lett. 102, 145002 (2009). [7] M. Borghesi et al., Fusion. Sci. Technol. 49, 412 (2006). [8] A. Andreev et al., Phys. Rev. Lett. 101, 155002 (2008). [9] A. Henig et al., Phys. Rev. Lett. 103, 245003 (2009). [10] A. P. L. Robinson et al., Plasma Phys. Control. Fusion 51, 024004 (2009). [11] A. P. L. Robinson et al., Plasma Phys. Control. Fusion 51, 095006 (2009). [12] A. Macchi et al., Phys. Rev. Lett. 103, 085003 (2005). [13] J. F. L. Simmons and C. R. McInnes, Am. J. Phys. 61, 2005 (1993). [14] F. Pegoraro et al., Phys. Rev. Lett. 99, 065002 (2007) [15] M. Geissler et al., New. J. Phys. 8, 186 (2006); M. Geissler et al., ibid. 9, 218 (2007). [16] S. S. Bulanov et al., Phys. Rev. E 78, 026412 (2008).
7
FIG. 1: Schematic profiles of ion (ni , black line) and electron (ne , blue line) densities and of the electrostatic field (Ez , red line) in the early stage of CP laser foil interactions, before ions move. (a)-(c) are respectively for the foils in thickness of semi-infinite, µm-scale, and nm-scale cases.
3.05
nip/nip0
0 5 3 3.02 3 3.05 3 z (µm) z (µm) 1
2 0
1
2 0
3.1 z (µm) 0.25
3.2
3
4 z (µm)
0.5 0
0.05
3
0 3.01 3.02
(g)
1
20
0
10
5
0 0.1
3
3.1 3.2 z (µm)
0
0
3
3
3.05
(h)
1
2
0
1
−1 0
0
0 30
e
3
0.5
1 100
z
100
5 −5 z (µm)
1
2 200
ne(nc)
x (µm) x (µm)
200
0
0
3 300 (f)
(e)
(d) t=80fs ne/ne0
E (m c! /e)
(c) t=34fs
−5
& nip (nc)
(a) t=12.5fs (b) t=20fs
3
4 z (µm)
5
−1
FIG. 2: (a)-(d) Electron ne /ne0 and proton nip /nip0 densities in the (z, x) plane at respectively t = 12.5, 20, 34 and 80fs for a hydrogen nanofoil with ne0 = nip0 = 200nc and l0 = 8nm irradiated by CP laser at I0 = 3.4 × 1019 W/cm2 . (e)-(h) are the corresponding longitudinal profiles of ne (blue), nip (black) and the electrostatic field Ez (red) cut at x = 0.
8
e
x (µm)
e0
10
n /n ic
ic0
0
5 −5
n /n ip
2
ip0
0
3 3.02 3 3.05 3 z (µm) z (µm)
3.1 z (µm)
3.2
3
3.5
4 z (µm)
1
1
10
1
2 0
1
2
0
0.25
0.5 0
3
3.01
0 2
2 10
(g)
3
(h)
3.05
0
1.5 1
1
1
0
0
0.5
10
0
0
−1
−0.5
−2
10
0.05
0.5 10
0
10 2.99
3 0
10
0
10
4.5
1
1
10
5
(f)
2 10
2
0
5 −5
x (µm)
(e)
n /n
ne (nc) & Z*nic(nc) & nip(nc)
x (µm)
(d) t= 80fs
(c) t= 34fs
−5
0.1
3.1 3.2 z(µm)
Ez(mec!0/e)
(a) t= 12.5fs (b) t= 20fs
10 −1
3
4 z(µm)
−1
FIG. 3: (a)-(d) Electron ne /ne0 , C 6+ ion nic /nic0 and proton nip /nip0 densities at t = 12.5, 20, 34 and 80fs for a CH nanofoil with ne0 = 200nc and l0 = 8nm irradiated by CP laser at I0 = 3.4× 1019 W/cm2 , where the density ratio of C6+ and H+ are nic0 : nip0 = 8 : 1; (e)-(h) the corresponding longitudinal profiles of ne (blue), nip (black), Z ∗ nic = 6nic (green) and Ez (red).
5 6
8 10 z (µm)
0
0.15
particles / MeV
0.3
(b)
8
10
+
H 6+
C
6
10
4
10
0
10 20 Energy (MeV/u)
x (µm)
1
0
0 5 22
0
20
20
40 60 t (2!/")
0
~t1/3
10 5 0
20
23
80
(d)
15
0
(e)
−5
(c)
40 60 t (2!/")
80
particles / MeV
nip(nc)
0
2 Ez(mec"/e)
(a)
peak Energy (MeV)
x (µm)
−5
6
24 z (µm)
25
0.1
(f)
0.2 +
H
10
C6+ 4
10
50 75 100 Energy (MeV/u)
FIG. 4: (a) Density map and (b) energy spectrum of the proton beam obtained at t = 130fs by the Leaky LS RPA in Fig. 3. (c) and (d) are respectively the peak Ez on and the peak energy of the proton beam varying with time t. (e) and (f) are density map and energy spectrum of 100MeV proton beams obtained at t = 240fs by CP lasers at intensities of 2 × 1020 cm2 , other parameters see the text. The dashed line in (b) is the energy spectrum for the single-species case in Fig. 2.
9
