How much laser power can propagate through fusion plasma?

Pavel Lushnikov1,2,3 and Harvey A. Rose3

1Landau Institute for Theoretical Physics 2Department of Mathematics, University of Notre Dame 3Theoretical Division, Los Alamos National Laboratory

D+T=4He (3.5 Mev)+n (14.1 Mev)

Thermonuclear burn

D+ 3He =4He (3.7 Mev)+p (14.7 Mev)

Required temperature: 10 KeV

Required temperature: 100 KeV3He + 3He =2p+4He (12.9 MeV)

Indirect Drive Approach to Fusion

Thermonuclear target

National Ignition Facility

National Ignition Facility Target Chamber

Target

192 laser beams

Laser pulse duration: 20 ns Total laser energy: 1.8 MJLaser Power: 500 TW

Goal: propagation of laser light in plasma with minimal distortion to produce x-rays in exactly desired positions

Difficulty : self-focusing of light

Nonlinear medium

Self-focusing of laser beam

Laser beam

Singularity point

z

- Nonlinear Schrödinger Eq.

- amplitude of light

Strong beam spray No spray

Laser propagation in plasma

Experiments (Niemann, et al , 2005) at the Omega laser facility (Laboratory for Laser

Energetics, Rochester)

Cross section of laser beam intensity after propagation through plasma Dashed circles correspond to beam width for propagation in vacuum.

No beam spray Beam spray

Plasma parameters at Rochester experiment

~2keVeT

14 20 1.5 10 W/cmI

Intensity threshold for beam spray

Electron temperature

/ 0.2e cn n Plasma Density 2

024

ec

mn

e

Plasma composition: plastic

Comparison of theoretical prediction with experiment

* 2 /i i i ii i

Z n Z n Z -effective plasma ionization number

2 2

204

e

c e e

nF I eI

n m T -dimensionless laser

intensity

in - number density for I-th ion species

iZ - ionization number for I-th ion species

- Landau damping

F - optic f-number

National Ignition Facility for He-H plasma

~5keVeT

* ~ 1ZThermal effects are negligible in contrast with Rochester experiments

Laser-plasma interactions

- amplitude of light

- low frequency plasma density fluctuation

- Landau damping- speed of sound

Thermal fluctuations

4/3 * 2/31 (7 ) ( )SH

eik Z

e

128v ,

3SH e ein

- thermal conductivity

ei - electron-ion mean free path

ev /ei ei -electron-ion collision rate

vosc eE me0 - electron oscillation speed

Thermal transport controls beam sprayas plasma ionization increases

Non-local thermal transport model first verified* at Trident (Los Alamos)

Large correlation time limit

- Nonlinear Schrödinger Eq.

Small correlation time limit

- light intensity is constant

Laser power and critical power

Power of each NIF’s 48 beams: P=8x1012 Watts

Critical power for self-focusing: Pcr=1.6x109 Watts

P/ Pcr =5000

3

2e c

cr ee

c T nP m

e n

Lens Random phase plate

Laser beamPlasma

- optic

Spatial and temporal incoherence of laser beam

“Top hat” model of NIF optics:

- optic

23

Intensity fluctuations fluctuate, in vacuum, on time scale Tc

Laser propagation direction, z

I E2

= intensity

Idea of spatial and temporal incoherence of laser beam is to suppress self-focusing

3D picture of intensity fluctuations

Fraction of power in speckles with intensity above critical per unit length

2 16 22 10 W/cmcr cI P F

15 2 -12x10 W/cm 0.8 cmscatterdI P

For NIF:

- amount of power lost for collapses per 1 cm of plasma

Temporal incoherence of laser beam

“Top hat” model of NIF optics:

- optic

Duration of collapse event collapse c s crT l c P P

c sl c - acoustic transit time across speckle

Condition for collapse to develop:

2

2 scollapse c cr c

c

cT T P P T

l

-probability of collapse

decreases with cT

Existing experiments can not be explainedbased on collapses. Collective effects dominate.

Cross section of laser beam intensity after propagation through plasma Dashed circles correspond to beam width for propagation in vacuum.

No beam spray Beam spray

Unexpected analytical result:Collective Brillouin instability

Even for very small correlation time, ,there is forward stimulated Brillouin instability

- light

- ion acoustic wave

Numerical confirmation: Intensity fluctuations power spectrum1

/ kmcs

k /

km

- acoustic resonance1P. M. Lushnikov, and H.A. Rose, Phys. Rev. Lett. 92, p. 255003 (2004).

Instability for

Random phase plate:

Wigner distribution function:

Eq: in terms of Wigner distribution function:

Boundary conditions:

Equation for density:

Fourier transform:

-closed Eq. for Wigner distribution function

Linearization:

Dispersion relation:

Top hat:

Instability growth rate:

2 2

0 204

e

c e e

nF I eI

n m T

Maximum of instability growth rate:

- close to resonance

and depend only on : 2 2

0 204

e

c e e

nF I eI

n m T

Absolute versus convective instability:

is real : convective instability only.

There is no exponential growth of perturbations in time – only with z.

Density response function:

- self energy

As

Pole of corresponds to dispersion relation above.

Collective stimulated Brillouin instability Versus instability of coherent beam:

- coherent beam instability

- incoherent beam instability

Instability criteria for collective Brillouin scattering

-convective growth rate

perturbations ~

Instability is controlled by the single parameter:

I 1

F2 ne

nc

vosc

ve

2

1

Pspeckle

Pcritical

2 2

0 204

e

c e e

nF I eI

n m T - dimensionless laser

intensity

- Landau damping

F - optic f-number

Comparison of theoretical prediction with experiment

* 2 /i i i ii i

Z n Z n Z -effective plasma ionization number

in - number density for I-th ion species

iZ - ionization number for I-th ion species

Solid black curve – instabilitythreshold

~2keVeT14 2

0 1.5 10 W/cmI

Second theoretical prediction:

Threshold for laser intensity propagation does not depend oncorrelation time for cT 1.s c cc l T

=3.4pscT

=1.7pscT

10.2pss cc l

National Ignition Facility for He-H plasma

~5keVeT

* 1.7Z Thermal effects are negligible in contrast with Rochester experiments

15 20 2x10 W/cmI

By accident(?) the parameters of the original NIF design correspond to the instability threshold

NIF:

Theoretical prediction for newly (2005) proposedNIF design of hohlraum with SiO2 foam:

~5 keVeT

He is added to a background SiO2 plasma, in order to increase the value of

and hence the beam spray onset intensity.

15 -10/ 0.1, 8, 5.4 10 sece cn n F

Fluctuations are almost Gaussian below threshold:

And they have non-Gaussian tails well above FSBS instability threshold:

Below threshold a quasi-equilibrium is attained:

True equilibrium can not be attained because slowly grows with z for any nonzero Tc:

Slope of growth can be found using a variantof weak turbulence theory:

Linear solution oscillate:

But is a slow function of z

Boundary value:

For small but finite correlation time, ,kinetic Eq. for Fk is given, after averaging over fastrandom temporal variations, by:

Solution of kinetic Eq. for small z:

which is in agreement with numerical calculation of

depends strongly on spectral formof , e.g. for Gaussan value of is about 3 times larger.

Change of spectrum of with propagation distance is responsible for change of the slope :

Growth of is responsible for deviationof beam propagation from the geometricaloptics approximation which could be critical for the target radiation symmetry in fusion experiments.

Intermediate regime near the threshold of FSBS instability

Electric field fluctuations are still almost Gaussian:

But grows very fast due to FSBS instability:

Key idea: in intermediate regime laser correlation length rapidly decreases with propagation distance:

Laser beam

Backscattered light

Plasma

and backscatter is suppressed due to decrease ofcorrelation length1

1H. A. Rose and D. F. DuBois, Phys. Rev. Lett. 72, 2883 (1994).

Light intensity:

Laser intensityinnintensityinte

Dopant concentration

Weak regime Intermediate regime Strong regime

Geometric optics Ray diffusion Beam spray

For example: 1% Xe added to He plasma, with temperature 5keV, ne/nc= 0.1, Lc=3m, 1/3m light, induces transition between weak and

intermediate regime for 70% of intensity compare with no dopant case.

Small amount (~ 1%) of high ionization state dopant may lead to significant thermal response, T, because

T ndopant Zdopant2

Zdopant - dopant ionization; ndopant – dopant concentration

< I > (W/cm2)

Xen

on (

Z =

40)

fra

ctio

nXenon dopant in He plasma:

- light

- ion acoustic wave

Backward Stimulated Brillouin instability:

Suggested explanation:

Nonlinear thermal effects ~ Z2

Result: change of threshold of FSBS due toChange in effective , and, respectively, changeOf threshold for backscatter.

April-May 2006: new experiments of LANL teamat Rochester: very high stimulated Raman scattering

- light

- Langmuir wave

Theoretical prediction: beam spray vs. stimulated Raman scattering

SRS intensity amplification in single hot spot

Probability density for hot spot intensity

Average amplification diverges

for

- amplification factor

Leads to enhanced (but not excessive) beam spray,

Add high Z dopant to increase thermal component of plasma response

Causing rapid decrease of laser correlation lengthwith beam propagation1

Raise backscatter intensity threshold2

Diminished backscatter

How to control beam propagation

2H. A. Rose and D. F. DuBois, PRL 72, 2883 (1994).1P. M. Lushnikov and H. A. Rose, PRL 92 , 255003 (2004).

Conclusion

Analytic theory of the forward stimulated Brillouin scattering (FSBS) instability of a spatially and temporally incoherent laser beam is developed. Significant self-focusing is possible even for very small correlation time.

In the stable regime, an analytic expression for the angular diffusion coefficient, , is obtained, which provides an essential corrections to a geometric optics approximations.

Decrease of correlation length near threshold of FSBScould be critical for backscatter instability and future operations of the National Ignition Facility.

D+T=4He (3.5 Mev)+n (14.1 Mev)

Thermonuclear burn

D+ 3He =4He (3.7 Mev)+p (14.7 Mev)

Required temperature: 10 KeV

Required temperature: 100 KeV3He + 3He =2p+4He (12.9 MeV)