How much laser power can propagate through fusion plasma?

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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
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Thermonuclear burn
DT4He (3.5 Mev)n (14.1 Mev)
Required temperature 10 KeV
D 3He 4He (3.7 Mev)p (14.7 Mev)
Required temperature 100 KeV
3He 3He 2p4He (12.9 MeV)
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Indirect Drive Approach to Fusion
Thermonuclear target
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National Ignition Facility
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National Ignition Facility Target Chamber
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Target
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192 laser beams
Laser pulse duration 20 ns Total laser energy
1.8 MJ Laser Power 500 TW
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Goal propagation of laser light in plasma with
minimal distortion to produce x-rays in exactly
desired positions
Difficulty self-focusing of light
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Self-focusing of laser beam
Nonlinear medium
Laser beam
z
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Strong beam spray No spray
Laser propagation in plasma
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Experiments (Niemann, et al , 2005) at the Omega
laser facility (Laboratory for Laser Energetics,
Rochester)
Beam spray
No beam spray

Cross section of laser beam intensity after
propagation through plasma Dashed circles
correspond to beam width for propagation in
vacuum.
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Plasma parameters at Rochester experiment
Electron temperature
Intensity threshold for beam spray
Plasma Density
Plasma composition plastic
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Comparison of theoretical prediction with
experiment

• dimensionless laser
• intensity

- Landau damping
- optic f-number
-effective plasma ionization number
- number density for I-th ion species
- ionization number for I-th ion species
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National Ignition Facility for He-H plasma
Thermal effects are negligible in contrast with
Rochester experiments
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Laser-plasma interactions
- amplitude of light
- low frequency plasma density fluctuation
- Landau damping
- speed of sound
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Thermal fluctuations
- thermal conductivity
- electron oscillation speed
- electron-ion mean free path
-electron-ion collision rate
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Thermal transport controls beam sprayas plasma
ionization increases
Non-local thermal transport model first verified
at Trident (Los Alamos)
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Large correlation time limit
- Nonlinear Schrödinger Eq.
Small correlation time limit
- light intensity is constant
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Laser power and critical power
Power of each NIFs 48 beams P8x1012 Watts
Critical power for self-focusing Pcr1.6x109
Watts
P/ Pcr 5000
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Laser beam
Plasma
Lens
Random phase plate
- optic
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Spatial and temporal incoherence
of laser beam
Top hat model of NIF optics
- optic
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Intensity fluctuations fluctuate, in vacuum, on
time scale Tc
Idea of spatial and temporal incoherence of
laser beam is to suppress self-focusing
Laser propagation direction, z
intensity
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3D picture of intensity fluctuations
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Fraction of power in speckles with intensity
above critical per unit length
For NIF

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

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Temporal incoherence of laser beam
Top hat model of NIF optics
- optic
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Duration of collapse event
- acoustic transit time across speckle
Condition for collapse to develop

• probability of collapse
• decreases with

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Existing experiments can not be explained based
on collapses. Collective effects dominate.
Beam spray
No beam spray

Cross section of laser beam intensity after
propagation through plasma Dashed circles
correspond to beam width for propagation in
vacuum.
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Unexpected analytical result Collective
Brillouin instability
Even for very small correlation time,
, there is forward stimulated Brillouin
instability
- light
- ion acoustic wave
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Numerical confirmation Intensity fluctuations
power spectrum1
k / km
w / kmcs
- acoustic resonance
1P. M. Lushnikov, and H.A. Rose, Phys. Rev. Lett.
92, p. 255003 (2004).
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Instability for
Random phase plate
Wigner distribution function
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Eq
in terms of Wigner distribution function
Boundary conditions
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Equation for density
Fourier transform
-closed Eq. for Wigner distribution function
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Linearization
Dispersion relation
Top hat
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Instability growth rate
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Maximum of instability growth rate
- close to resonance
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and depend only on

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Absolute versus convective instability
is real convective instability only.
There is no exponential growth of perturbations
in time only with z.
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Density response function
- self energy
Pole of corresponds to dispersion
relation above.
As
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Collective stimulated Brillouin instability
Versus instability of coherent beam
- coherent beam instability
- incoherent beam instability
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Instability criteria for collective Brillouin
scattering
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-convective growth rate
perturbations
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Instability is controlled by the single
parameter

• dimensionless laser
• intensity

- Landau damping
- optic f-number
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Comparison of theoretical prediction with
experiment
Solid black curve instability threshold
-effective plasma ionization number
- number density for I-th ion species
- ionization number for I-th ion species
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Second theoretical prediction
Threshold for laser intensity propagation does
not depend on correlation time for
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National Ignition Facility for He-H plasma
Thermal effects are negligible in contrast with
Rochester experiments
NIF
By accident(?) the parameters of the original
NIF design correspond to the instability
threshold
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Theoretical prediction for newly (2005)
proposed NIF design of hohlraum with SiO2 foam
He is added to a background SiO2 plasma, in
order to increase the value of n and hence the
beam spray onset intensity.
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Fluctuations are almost Gaussian below threshold
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And they have non-Gaussian tails well above FSBS
instability threshold
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Below threshold a quasi-equilibrium is attained
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True equilibrium can not be attained because
slowly grows with z for any nonzero Tc
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Slope of growth can be found using a
variant of weak turbulence theory
Linear solution oscillate
But is a slow function of z
Boundary value
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For small but finite correlation time,
, kinetic Eq. for Fk is given,
after averaging over fast random temporal
variations, by
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Solution of kinetic Eq. for small z
which is in agreement with numerical calculation
of
depends strongly on spectral form of ,
e.g. for Gaussan value of is about 3
times larger.
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Change of spectrum of with propagation
distance is responsible for change of the slope

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

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Intermediate regime near the threshold of FSBS
instability
Electric field fluctuations are still almost
Gaussian
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But grows very fast due to FSBS
instability
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Key idea in intermediate regime laser
correlation length rapidly decreases with
propagation distance
Plasma
Laser beam
Backscattered light
Light intensity
and backscatter is suppressed due to decrease
of correlation length1
1H. A. Rose and D. F. DuBois, Phys. Rev. Lett.
72, 2883 (1994).
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Small amount ( 1) of high ionization state
dopant may lead to significant thermal response,
dT, because
Zdopant - dopant ionization ndopant 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, Lc3mm, 1/3mm
light, induces transition between weak and
intermediate regime for 70 of intensity compare
with no dopant case.
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Xenon dopant in He plasma
Xenon (Z 40) fraction
lt I gt (W/cm2)
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Backward Stimulated Brillouin instability
?
- light
- ion acoustic wave
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Suggested explanation
Nonlinear thermal effects Z2
Result change of threshold of FSBS due to Change
in effective , and, respectively, change Of
threshold for backscatter.
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April-May 2006 new experiments of LANL team at
Rochester very high stimulated Raman scattering
- light
- Langmuir wave
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Theoretical prediction beam spray vs.
stimulated Raman scattering
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SRS intensity amplification in single hot spot
- amplification factor
Probability density for hot spot intensity
Average amplification
diverges for
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How to control beam propagation
Add high Z dopant to increase thermal component
of plasma response
Leads to enhanced (but not excessive) beam spray,
Causing rapid decrease of laser correlation
length with beam propagation1
Raise backscatter intensity threshold2
Diminished backscatter
2H. A. Rose and D. F. DuBois, PRL 72, 2883 (1994).
1P. M. Lushnikov and H. A. Rose, PRL 92 , 255003
(2004).
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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
  FSBS
• could be critical for backscatter instability and
  future operations
• of the National Ignition Facility.

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Thermonuclear burn
DT4He (3.5 Mev)n (14.1 Mev)
Required temperature 10 KeV
D 3He 4He (3.7 Mev)p (14.7 Mev)
Required temperature 100 KeV
3He 3He 2p4He (12.9 MeV)