underdense pasma.1 To check these predictions we have developed a relativistic, two-dimensional particle-in-cell code. Test simulations that validate the code ...
Particle-in-Cell Simulations of Particle Acceleration E. A. Startsev and C. J. McKinstrie Laboratory for Laser Energetics, U. of Rochester Previous analytical calculations suggest that a preaccelerated test particle can be ponderomotively accelerated to energies in excess of 100 MeV by an intense electromagnetic pulse propagating in a underdense plasma.1 To check these predictions we have developed a relativistic, two-dimensional particle-in-cell code. Test simulations that validate the code will be described in detail. Preliminary simulations of the interaction of a preaccelerated electron bunch with an electromagnetic pulse in a plasma will be presented. This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC03-92SF19460, the University of Rochester, and the New York State Energy Research and Development Authority. 1. C. J. McKinstrie and E. A. Startsev, Phys. Rev. E 54, R1070 (1996); 56, 2130 (1997).
Particle-in-Cell Simulations of Particle Acceleration E. A. STARTSEV and C. J. McKINSTRIE University of Rochester Laboratory for Laser Energetics
Abstract
Previous analytical calculations suggest that a preaccelerated test particle can be ponderomotively accelerated to energies in excess of 100 MeV by an intense electromagnetic pulse propagating in an underdense pasma.1 To check these predictions we have developed a relativistic, two-dimensional particle-in-cell code. Test simulations that validate the code will be described in detail. Preliminary simulations of the interaction of a preaccelerated electron bunch with an electromagnetic pulse in a plasma will be presented.
1. C. J. McKinstrie and E. A. Startsev. Phys. Rev. E 54, R1070 (1996); 56, 2130 (1997)
An intense laser pulse in a plasma repels the particle
• In vacuum the pulse displaces the particle but leaves it at rest. A
A
• In a plasma the pulse repels the particle if intensity exceeds a threshold
(
VB = Vp
)
2 a2B = 2 γ pγ A − upuA − 1 .
Vp
where aB = e|A|mec2
VA
(
• Here, γ g = 1 1− v 2g P1984
VC
)
12
(
)
and up = γ 2p − 1
12
.
The particle can be effectively accelerated
(
)
• The energy gain δγ = 2 u2p γ A − γ p up uA . Threshold and gain
1000 γp = 30 100
Energy gain Repelling threshold
10 1 0.1 0
5
10
15
20
25
30
Injection energy (× mec2) • With energy measured in units of the particle rest mass, if the injection energy γA = 7.0, the repelling threshold a2B = 8.3 and the energy gain δγ = 120. P1985
1-D simulations show good agreement with theory1, 2 • 1-D propagation of an intense circularly polarized laser pulse in a plasma is described by the equations
] (1− v2g), φ ′′ = (2π)2[1− vg • (1+ φ) ψ ], [
ux = vg • (1+ φ) − ψ
[(
) (
)(
ψ = 1+ φ 2 − 1− v2g • 1+ a2
)]
12
,
where φ = eϕ/mc2 is electrostatic potential, vg is pulse group speed, and a = eA/mc2. • The fluid and PIC results were compared for a = 5 • sin(πx/9), ωp/ω0 = 0.1.
Ux
10
0
5
–0.1
eE x mcω 0
0 –5 –10
–0.3 –0.4
2
4
6 X/λ0
P2009
–0.2
8
10
–0.5
2
4
6
8
10
X/λ0 2. P. K. Kaw, A. Sen and T. Katsouleas, Phys. Rev. Lett. 68, 3172 (1992).
Forward SRS was observed
t = 1600 λ0
t = 1000 λ0
• At t = 1600 λ0 the nonlinear state of the beam evolution leads to spectral cascading3.
3C.
D. Decker, W. B. Mori, and T. Katsouleas, Phys. Rev. E, 50, R3338 (1994). P2041
–1 0.02
–1 0.05
eEz/mω0c
1
eEx/mω0c
• At t = 1000 λ0 RFS grows at the head of the pulse from anomalously large noise source provided by RBS.
1
–0.02
0
x/λ0
450
–0.05
0
x/λ0
450
104
104
eEz/mω0c
• The 1-D simulation with a0 = 0.8, ω0/ωp = 10, l = 200 λ0, λ0 = 1 µm, and linear polarization.
100 0
10 k/kp
20
100
0
10 k/kp
20
Multistage ponderomotive acceleration is possible in a plasma with a density gradient Vpi i+1 VA
n (x ) n0
Vpi+ 1
i VA
i+2 VA i+1 VA
Vp = 1 −
2
i i +1 i γp • On each stage γ A = γ A i , where i γA
denotes quantities before acceleration and i + 1 denotes quantities after acceleration. • Also, the threshold for each stage is i 1 γ p γ iA 1+ = i + i . 2 2γ γp A
a2i
• If, on each stage, γ ip γ iA = α = const, then ath i = const. and the final energy after N stages is γ final = α 2N • γ 0A . A P2026
1 n (x ) 2 ncr
• If the density profile is chosen as n(x) = n0/(1 + x/l)2, then γ ip+1
γ ip
=α
2
and
γ ip+ 1 γ iA+ 1
γ ip
= i = α = const. γA
• On the distance L, these will be N=
1 ln (1+ L l) number of stages; 2 ln γ 0p γ 0
(
A
)
here, γ 0p = ω 0 ωp . • If L/l = 1000, γ 0 = 5, ω /ω = 10, then N = 5 0 p A and γ final = 5000 me•ci = 2.5 GeV! A
Accelerated particles were observed t = 3000 λ0
t = 1000 λ0
• At the time t ~ γp2 • l ~ 3000 λ0, preinjected particles have completed their acceleration and are moving ahead of the pulse.
eEx/mω0c
eEx/mω0c
–0.02 3
–0.02 10
eEz/mω0c
• Pulse intensity a = 10 and plasma density γp = ω0/ωp = 30, l = 3 λ0. Pulse was circularly polarized.
0.10
0.3
eEz/mω0c
• A bunch of electrons was preinjected ahead of the pulse in the direction of the pulse propagation with initial energies γ0 = 7.
2
–8 0
x/λ0
9
x/λ0
9
10
15
Pz
Pz
–10 –20
• Energy gain δγ = (γp /γ0)2 • γ0 ~ 130 was observed.
Px
–8
140
140
0
x/λ0
200
200
Px
Px
–20 P2042
0
0
x/λo
9
0
0
Px
9
Relativistic self-focusing was observed for a pulse with P > Pc a0 = 0.75 43.0
y/λ0
0.78
y/λ0 –0.062
a0 = 0.35
e|Ez|/mω0c
e|Ez|/mω0c
eEx/mω0c
eEx/mω0c
21.5
• The 2-D simulations were made with the parameters (a) a0 = 0.75, b) a0 = 0.35, ω0/ωp = 5, l = 40 λ0, r⊥ = 7 λ0, λ0 = 1 µm.
0 0.023
• For these parameters LR = π • r⊥2 /λ0 = 150 • λ0, (a) P/Pc = 2, (b) P/Pc = 0.5.
• In the case of P/Pc = 2 beam stays focused for two Rayleigh lengths. –0.022
0.0 0
100 0 X/λ0
5
100 X/λ0 eEz/mω0c
ky/kp
eEz/mω0c
0
5 kx/kp
P2043
0.25
21.5
0.0 43.0
0 0.062
t = 300 λ0
10 0
5 kx/kp
10
• Substantial Raman sidescatter (RSS) was present. • RSS has only a downshifted Stokes component because the anti-Stokes cannot be simultaneously resonant. • In comparison, for (b) P/Pc = 0.5, no self focusing was observed.
Electrostatic field does not prevent particle acceleration t = 100 • λ0
y/λ0
80
e|Ez|/mω0c
y/λ0
5
e|Ez|/mω0c
40
0 80
eEx/mω0c
0 0.082
eEx/mω0c
n/n0
–0.025 15
n/n0
40
0
0 0.0
4.5
9.0 0.0
4.5
x/λ0
Px
8 0
P2044
LR/λ0 = π (r⊥/λ0)2 ≈ 300.
• Though the electrostatic field is significant, it is negligible ahead of the pulse where particles are accelerated. • The initial particle energy was γ0 = 5. The energy gain δγ = 25 was observed. • Particles are spread in transverse direction by the ponderomotive force in the angle
31
16
–8
9.0
x/λ0
24 Px
• 2-D simulation of the particle acceleration was conducted with parameters a0 = 10, ω0/ωp = 10, r⊥/λ0 = 10, l/λ0 = 3.
• The power of the pulse is being quickly depleted by the wake generation.
40
0 80
y/λ0
t = 400 • λ0
23 15 7
0
9 x/λ0
–1
9
0 x/λ0
θ=
∆pmax ⊥ γ max
= 0.01
Summary/Conclusion
• A simple 1-D acceleration scheme was suggested that uses the PM force to accelerate preinjected electrons to energies as high as 100 MeV. • The model of the multistage pondermotive acceleration in a plasma with a density profile predicts energy gains of δγ . 2 GeV on a distance of several centimeters. • This scheme relies on the fact that the group speed of the laser pulse is less than c. • A 2-D particle-in-cell code was developed to check this prediction. • Test simulations that validate the code were presented. • Simulations of the interaction of a preaccelerated electron bunch with an electromagnetic pulse in a plasma show evidence of electron acceleration.
P1990
