How much laser power can propagate through fusion plasma? Pavel Lushnikov 1,2,3 and Harvey A. Rose 3 1 Landau Institute for Theoretical Physics 2 Department of Mathematics, University of Notre Dame 3 Theoretical Division, Los Alamos National Laboratory

D+T= 4 He (3.5 Mev)+n (14.1 Mev) Thermonuclear burn D+ 3 He = 4 He (3.7 Mev)+p (14.7 Mev) Required temperature: 10 KeV Required temperature: 100 KeV 3 He + 3 He =2p+ 4 He (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 MJ Laser 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 Schr ö dinger 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 Intensity threshold for beam spray Electron temperature Plasma Density Plasma composition: plastic

Comparison of theoretical prediction with experiment -effective plasma ionization number -dimensionless laser intensity - number density for I-th ion species - ionization number for I-th ion species - Landau damping - optic f-number

National Ignition Facility for He-H plasma Thermal 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 - thermal conductivity - electron-ion mean free path -electron-ion collision rate - electron oscillation speed

Thermal transport controls beam spray as plasma ionization increases Non-local thermal transport model first verified * at Trident (Los Alamos)

Large correlation time limit - Nonlinear Schr ö dinger Eq. Small correlation time limit - light intensity is constant

Laser power and critical power Power of each NIF’s 48 beams: P=8x10 12 Watts Critical power for self-focusing: P cr =1.6x10 9 Watts P/ P cr =5000

LensRandom phase plate Laser beam Plasma - optic

Spatial and temporal incoherence of laser beam “Top hat” model of NIF optics: - optic

23 Intensity fluctuations fluctuate, in vacuum, on time scale T c Laser propagation direction, z = 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 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 - acoustic transit time across speckle Condition for collapse to develop: -probability of collapse decreases with

Existing experiments can not be explained based 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 spectrum 1  / k m c s k / k m - acoustic resonance 1 P. M. Lushnikov, and H.A. Rose, Phys. Rev. Lett. 92, p (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:

Maximum of instability growth rate: - close to resonance

and depend only on :

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: - dimensionless laser intensity - Landau damping - optic f-number

Comparison of theoretical prediction with experiment -effective plasma ionization number - number density for I-th ion species - ionization number for I-th ion species Solid black curve – instability threshold

Second theoretical prediction: Threshold for laser intensity propagation does not depend on correlation time for

National Ignition Facility for He-H plasma Thermal effects are negligible in contrast with Rochester experiments By accident(?) the parameters of the original NIF design correspond to the instability threshold NIF:

Theoretical prediction for newly (2005) proposed NIF design of hohlraum with SiO 2 foam: He is added to a background SiO 2 plasma, in order to increase the value of and hence the beam spray onset intensity.

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 T c :

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:

For small but finite correlation time,, kinetic Eq. for F k is given, after averaging over fast random temporal variations, by:

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.

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

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.

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 of correlation length 1 1 H. A. Rose and D. F. DuBois, Phys. Rev. Lett. 72, 2883 (1994). Light intensity:

Weak regime Intermediate regime Strong regime Geometric optics Ray diffusionBeam spray For example: 1% Xe added to He plasma, with temperature 5keV, n e /n c = 0.1, L c =3  m, 1/3  m 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 Z dopant - dopant ionization; n dopant – dopant concentration

(W/cm 2 ) Xenon (Z = 40) fraction Xenon dopant in He plasma:

- light - ion acoustic wave Backward Stimulated Brillouin instability: 

Suggested explanation: Nonlinear thermal effects ~ Z 2 Result: change of threshold of FSBS due to Change in effective, and, respectively, change Of threshold for backscatter.

April-May 2006: new experiments of LANL team at 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 length with beam propagation 1 Raise backscatter intensity threshold 2 Diminished backscatter How to control beam propagation 2 H. A. Rose and D. F. DuBois, PRL 72, 2883 (1994). 1 P. M. Lushnikov and H. A. Rose, PRL 92, (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 FSBS could be critical for backscatter instability and future operations of the National Ignition Facility.

D+T= 4 He (3.5 Mev)+n (14.1 Mev) Thermonuclear burn D+ 3 He = 4 He (3.7 Mev)+p (14.7 Mev) Required temperature: 10 KeV Required temperature: 100 KeV 3 He + 3 He =2p+ 4 He (12.9 MeV)