Growth, Saturation, and Energetic Electron Generation in Two-Plasmon Decay Instabilities in

Direct-Drive Inertial Confinement Fusion 直接驱动惯性约束聚变中的双等离子体波衰变的增长，饱

和，和热电子的产生

闫锐University of Rochester

Inertial Confinement Fusion (ICF) aims to make fusion a clean and unlimited energy source

Laser heating pellet surface rocket liftoff of ablator -> inward shock wave

A hot spot is created in the dense core and reaction starts

~1 mm

• The National Ignition Facility (NIF) is aiming to achieve ignition within this year

Current ICF schemes

• 1-step

– Direct-drive compression

– Indirect-drive compression

• Multi-step

– Fast ignition

- Channeling (underdense) and/or hole boring (overdense)

- Cone

– Shock ignition

http://www.lle.rochester.edu

1. R. Betti et al.,Phys. Rev. Lett., 98, 155001 (2007)

A shock ignition pulse1

The two-plasmon-decay (TPD) instability is a laser-plasma-interaction (LPI) process in underdense plasma

• The matching condition

– k0=k1+k2

– ω0=ωp1+ωp2

• Since ωp1,ωp2 are close to ωpe,

TPD can only take place at n~1/4 ncr.

Laser

Plasma wave 1

Plasma wave 20 0,k

1 1, pk

2 2, pk

14

TPD Threshold*

( / ) / 81.86 1m m keVI L T

*Simon et al., Phys. Fluids 26, 3107 (1983)

• Effective compression of ICF targets requires fuel shells in low adiabatic states during implosion.

• Energetic (hot) electrons generated from laser-plasma interactions can preheat the shell and degrade the implosion

• The two-plasmon decay instability is a significant concern as a hot electron source.

Two-plasmon decay (TPD) is a potential danger for direct drive ICF

• Linear regime

• Quasi-steady state

Outline

Long-time-scale PIC simulations with OSIRIS1 have been performed for a range of parameters

laser

x

y

0

0

14 28 10 /I W cm

150

3e

L m

T keV

• Plane wave and Gaussian beams are used with the peak intensity range: 6e14-2e15W/cm2

• The simulation box is transversely periodic and thermal bath is applied at longitudinal boundaries.

• Diagnostics on the thermal-bath boundaries record the energy loss through the boundaries and the distribution of electrons having reached the boundaries.

1, R. A. Fonseca, L. O. Silva, F. S. Tsung et al., Lect Notes Comput Sc 2331, 342 (2002)

Simulation configuration for OMEGA parameters

A linear fluid code has been developed to study the linear regime of TPD

2

0

0

2

3

( )

4

e p

pp

p

v nev

t m nn

v n nt

en

0

0

0

| | 0

| | 0

| | 0

x x L

x x L

p p Ln n

BC's

v

E

• The fluid code solves the linear equations:

• Linear regime

– Threshold

– Growth rate

– Convective gain

L=150μm, T=2keV, I=1e15 W/cm2 Mi/me=3410

(OMEGA relevant parameters)

0 0.5 1 1.5 20

0.0005

0.001

0.0015PICSimon et al (SSWD) large-Afeyan&Williams (AW)

0k (ω /c)

0γ(ω )

A broad spectrum of plasma waves is observed in PIC simulations

• The reduction of the PIC growth rate in the low k┴ region is due to pump

depletion.

• The spatial location where the high k┴ modes grow is in agreement with

the homogeneous theory.

12

1, A. Simon, R. W. Short, E. A. Williams, and T. Dewandre, Phys.Fluids, 26, 3107 (1983)

2, B. B. Afeyan, E. A. Williams, Phys. Plasmas 4, 3827 (1997)

Fluid simulations are used to determine the nature of those high k┴ modes

2

0

0

2

3

( )

4

e p

pp

p

v nev

t m nn

v n nt

en

A similar spectrum to the PIC simulations was observed in the fluid simulations

Fluid

PIC

High k┴ modes can be identified as convective modes by fluid simulations

• Absolute modes can grow exponentially without limit in a linear system.

• Convective modes can only grow exponentially at early time and then saturate (separate).

x (c/0)

np/n

cr

1200 1400 1600 180010-8

10-7

10-6

10-5

07425.7 /t 012376./t

04950.5 /t

09901.0 /t

00.75( / )k c

00.18( / )k c

The convective gains observed in the fluid simulations are in agreement with the three-wave theory results

• The TPD equations can be cast into

the standard 3-wave form.1

• The order and trend of convective

modes’ amplification can be

estimated by eπΛ .

• The convective gain isn’t sensitive to k┴. This explains the broad

spectrum in PIC simulations

2

2

* ' / 21 1 1 2

* ' / 22 2 2 1

[ ] ( , ) ( , )

[ ] ( , ) ( , )

i xt x

i xt x

v a x t a x t e

v a x t a x t e

2

1 2'v v

1'L

1, M. N. Rosenbluth, R. B. White, and C. S. Liu, Phys. Rev. Lett., 31, 1190 (1973)

2keV keV

14 μm μm keV

πΛ 2.15 (1-0.00881T -0.0470T k )

(I λ L /T )/81.86

0.5 0.6 0.7 0.8 0.9 1101

102

103

Amplification (fluid)e (Rosenbluth Gain)

0k (ω /c)

0| | /k kn n

Amplification ~ exp[πΛ]

The high k┴ modes are energetically important in both linear and non-linear stages in the large-L PIC simulation

k (0/c)<

Ex2>

-2 -1 0 1 2

0

0.0005

Time (1/0)

Ex

en

erg

y(A

.U.)

2500 5000

100

101

• The linear growth of TPD is interrupted by ion density fluctuations

• The convective modes are saturated before reaching the convective gain because the initial noise has only been amplified by ~30 in amplitude

0141.4 /t

05656 /t

06363/t

Initial noise level X100

The convective modes are relatively more important in higher η cases

05656 /t

E1 0( / )mc eE1 0( / )mc e

05656 /t

I=1e15 W/cm2 ,L=150μm, T=2keV

η=3

I=8e14 W/cm2 ,L=150μm, T=3keV

η=1.6

• Quasi-steady state

– Energetic electron generation

– Collision and speckle effects

Net particle energy flux reaches a quasi-steady state after ~5ps

I=

In the quasi-steady state,

– Absorbed laser energy is balanced by the energy flux exiting the box

– The particle and field energies in the simulation box are essentially constant

t (ps)

Net particle energy flux

Ene

rgy

flux/

Lase

r flu

x

1 2 3 4 5 6 7 8 9 10 110

0.05

0.1

0.15

0.2

0.25

0.3

Left boundary

Right boundary

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10 12

time(ps)

Ex

ener

gy(

Arb

. Un

it)

Most hot electrons are produced in the nonlinear stage

L=150μmTe=3keVI=6X1014W/cm2

Electron >50keV distribution in px-py space

linear

saturation

quasi-steady state

Py(

mec

)

Px(mec) Px(mec)

The net energy flux exiting the right boundary includes significant contribution from the hot electrons

0-5keV 5-10keV 10-25keV

25-50keV

50-100keV

100-150keV

150-200keV

200-250keV

250-300keV

300-350keV

350-400keV

over 400keV

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

-1.53%

-0.41%

6.14%

3.10%

4.54%

5.49%

4.27%

1.97%

0.58%

0.19% 0.10% 0.24%Ene

rgy

flux/

Lase

r flu

x

Normalized instant net e- energy flux at t=9.9ps

Electron Energy

Electrons trapped in a plasma wave

Φe-

e-

Phase velocity

Electrons could be trapped in a plasma wave if their velocities are close to the phase velocity

kVph /

It is difficult to accelerate thermal electrons in a high-phase-velocity plasma wave

• Only the electrons with velocities close to Vph can

be coupled by the plasma wave

• These electrons are located at the tail of distribution function and the amount is small

f

v

eph vkV /

Electron distribution

The hot electrons are generated through staged acceleration initiated by new TPD modes with low phase velocity in the nonlinear stage

Ex in kx-x space (A. U.)

e-(>50keV) phase space

forward¼ critical surface

Ion density fluctuations are driven by plasma waves propagating to lower density regions

• The region of ion density fluctuations is spreading at the group velocity of plasma waves with the largest k.

• Ion fluctuations at the low density region can induce new TPD modes locally.

Vg=0.013c

n0 (nc)

t (p

s)

Ex energy

<E

x2>

(A

. U

.)

t (p

s)

Density fluctuation

δn (

n c)

Fluid simulations are performed to study the new modes in the nonlinear stage

2

0

0

2

3

( )

4

e p

pp

p

v nev

t m nn

v n nt

en

0

0

0

| | 0

| | 0

| | 0

x x L

x x L

p p Ln n

BC's

v

E

• The fluid code solves the linear TPD equations

• The density fluctuation is modeled by a static

n = n0(x) + δn

Fluid simulations produce modes similar to PIC simulations

• The background density profile is read from OSIRIS simulation results

• The spectrum obtained in the fluid simulation is similar to that from the PIC simulation

• The differences in the relative amplitudes may be due to e- acceleration.

Fluid PIC

I14max T(keV)/Ti(keV)

L η* Total Abs. Hot (>50kev) electrons

3 3/1.5 150 0.6 ~0 ~0

6 3/1.5 150 1.2 42% 17%

8 3/1.5 150 1.4 52% 24%

6 (collisional) 3/1.5 150 1.2 33% 5.5%

86.81/)/(

Threshold TPD

14 keVmm TLI *Simon et al., Phys. Fluids 26, 3107 (1983)

Electron-ion collision significantly reduces hot electron production in PIC simulations

-0.05

0

0.05

0.1

0.15

collisionlesscollisional

17%hot

5.5%hot

Using high-Z materials as the ablator can increasethe e-i collision and reduce the hot e- production

Laser speckle can also reduce the hot electron generation

• In experiments, polarization smoothing changes laser polarization even within a single speckle, which can reduce the region for electron acceleration

• Simulation with a narrow beam has shown a reduced hot electron generation

I14max T(keV)/Ti(keV)

L η* Total Abs.

Hot (>50kev) electrons

6 3/1.5 150 1.2 42% 17%

8 (W=4μm) 3/1.5 150 1.4 22% 5%

• Broad spectra of plasma waves are observed in PIC and fluid simulations

– The high- k┴ modes are identified as convective modes by fluid simulations. They

can become energetically dominant and cause pump depletion in high η cases

– A convective gain formula retaining the dependence on Te and k┴ is obtained

• In PIC simulations, significant laser absorption and hot electron generation occur in

the nonlinear stage

– Generation of hot electrons is correlated with new TPD modes correlated with ion

density fluctuations in the lower density region in the nonlinear stage

– Hot electrons are accelerated from the low density region to the high density

region through a staged process

– Both collision and speckle can reduce the hot electron generation

Summary