Laser Produced PlasmasWalter D. Zacherl

Applied Physics

References

[1] Richard Dendy. Plasma Physics; An Introductory Course. Cambridge University Press, New York, 1993.

Outline• Inertial Confinement Fusion

• X-ray Lasers

• Particle Accelerators

• Unusual States of Matter

• Laser Interactions with Matter

• Pondermotive Force

• Absorption

• Transportation

• Laser-driven Instabilities

• Rayleigh-Taylor Instability

Introduction•Research motivation.

•Generation of electricity through thermonuclear fusion.

•Nuclear weapons research.

•50 kJ laser, ionize ~10^22 atoms.

•10cm cube of air.

•3mm cube solid.

•Typical experiment.

•TW laser focused on a solid target creating a plasma 0.1-1mm.

•Plasma heated to keV temperatures and ablates at the speed of sound.

•~10 Mbar max thermal pressure (greater than conventional explosives.

•Pulse duration varies depending on application (ns – 100 fs).

Laser

Plasma PlumeS

olid

Inertial Confinement Fusion (ICF)•Commercial power generation using controlled thermonuclear explosions.

•D and T heated to 10 keV. Sufficient to penetrate the Coulomb barrier.

•Lawson criteria, n > 1020 s m-3.

•No confinement. Gas free to expand under its own pressure.

•Confinement time set by free expansion rate.

•Fusion rate 50% after radius expands by 25%.

~ 0.25R/cs where R is the minimum radius and is cs the speed of sound.

•cs at 10 keV ~ 106 m s-1.

D-T 10 keV

Inertial Confinement Fusion (ICF)•Using and replacing n by the Lawson criteria becomes R > 0.2 g cm-2.

•Lawson criteria could be satisfied by heating a 1cm sphere of D-T to 10 keV.

•Et = R3 4.8*109 J = (/0.2 g cm-2) GJ.

•Power in 300 times greater using efficiency assumptions, 10% laser, 10% D-T pellet, 30% power conversion.

•ETN = (/0.2 g cm-2) 0.3 TJ. 1 kton TNT = 4 TJ.

•Must increase the density.

•Pellet R = 3 g cm-2, central region 0.3 g cm-2.

•Compromise between energy, laser expense, and ability for the reactor vessel to contain the explosion puts the density at 300 g cm-3 (~1000 times solid density).

•As of 1993, Hollow shell targets have been compressed to 500-1000 times initial density but with temperatures below 10 keV.

= 300 g cm-3 and temp 10 keV, pressure = 106 Mbar.

•Pressure 104 times larger than directly irradiating a solid target. Pressure gap closed by target design.

Inertial Confinement Fusion (ICF)•Required convergence factor, = Rinit/Rfinal (ratio of initial to final radius).

•If a constant laser drive pressure Pd is applied, then PdV work done is

Ed = 4R2PddR 4/3 PdRinit3 (Rinit >> Rfinal).

Ef = 4/3 Rfinal3 3/2 Pf (ideal gas E = 3/2nkT, P = nkT).

Assuming Ed = Ef.

Rinitial/Rfinal = (3*Pf/2Pd)1/3.

•If Pf = 106 Mbar and Pd = 50 Mbar, the is 30. Volume compression is 3*104 which is sufficient to achieve 1000 times the solid density.

•Compression and target must be highly uniform to be compressed evenly.

•Thin shells are subject to the Rayleigh-Taylor instabilities.

X-ray Lasers•Plasmas used as lasing medium.

•Suitable atomic states have a very short lifetime. Power emitted by spontaneous decay is very large. Pump power must be larger.

•First example in the mid 1980s. ~50 eV photons, 20.6 nm. Population inversion produced in selenium by collisional excitation.

•Alternate scheme. Medium may be pumped by a separate hot plasma similar in concept to a flashlamp pumped laser. Difficult to match pumping photons and energy gap in lasing plasma.

•Recombination laser. Thin carbon fiber irradiated by an optical laser. The fiber is heated, ionized and ablates. As it cools, recombination occurs and electrons cascade down through the atomic states. Inversions occur at bottlenecks. A spontaneous photon can be amplified as it passes along the wire.

Particle Accelerators•E-fields in RF cavities limited by electric breakdown to ~ 108 V/m-1 necessitating the construction of very large accelerators.

•Construct an “accelerating cavity” in a structure that is already broken down.

•E-fields of 30 GeV/m-1 possible.

•The relativistic particle must experience a E-field that is constant to be accelerated. The rapidly oscillating traverse E-field of a single laser pulse, though large (1012 V/m-1), cannot be used to accelerate particles.

•Laser Wakefield Accelerator (LWA).

•Excite an electron plasma wave (Langmuir wave) with a high power laser.

•Laser pulse length l is matched to half the Langmuir oscillation period.

•Front of laser pulse gives each electron a push in the direction of propagation. The back gives it a push in the opposite direction. This drives the Langmuir oscillation.

•Oscillation continues after the pulse passes through the medium.

•Trailing electron beam is accelerated by the oscillation.

•The oscillatory wakes move at the group velocity of the pulse, slightly less than c.

•Requires a laser which delivers tens of Joules in ~ 1ps.

Unusual States of Matter•Laser-produced plasma are hot and dense.

•Electrostatic energy maybe comparable with thermal energy.

(Ze)2/(40r) ~ kT

•Particle motion strongly influenced by nearest neighbor. Exhibit crystalline behavior.

•Large-angle scattering becomes more important.

•Debye shielding length comparable to interparticle distance.

•Coulomb logarithm small, possibly negative if calculated in normal way.

•Structure of partially ionized ions affected by fields of surrounding particles.

•Spectroscopic line shifts.

•Smearing out of higher atomic states.

•Blackbody radiation pressure can become significant.

Pondermotive Force•Laser beam momentum transferred to plasma via the ponderomotive force.

•EOM for an e- in an EM field of frequency and wave number k traveling in the x-direction with an E-field of amplitude E0 in the y-direction and magnetic field B0 in the z-direction is

)cos(0 kxtm

eE

dt

dv

e

y )cos(0 kxtm

eBv

dt

dv

ey

x

•Although the fields are oscillatory, they exert a time averaged force if non-uniform.

•Assume low laser intensity and slowly varying fields.

•To first order, only the E-field is significant (B-field effect O(v/c)).

)cos(0 kxtm

eE

dt

dv

e

y

•Integration yields.

)sin( 0kxtvv oscy )cos( 0kxtv

y osc e

osc m

eEv 0

•The electron oscillates about its rest position.

Pondermotive Force•Add second order term.

)cos()( 00

0 kxtdy

dEyE

m

e

dt

dv

e

y

•Substituting the previous expression for y.

)(cos)cos( 020

000 kxt

dy

dE

m

evkxtE

m

eE

dt

dv

e

osc

e

y

•First term averages to 0. Second term averages to

dy

Ed

m

e

e

)2()(

2

1 202

•Force per unit volume acting on e-

dy

dU

n

n

dy

Ed

dt

dvmnF e

c

epyee 2

1)2()(

2

1 2002

p is the plasma frequency (p2= nee2/0me), nc is the critical density ( = p), and U = ½

0E02 is the max energy density of the EM wave.

•Similar expressions can be developed for the x and z directions.

Pondermotive Force

•The second term allows direct comparison to pressure.

2

4

1

2

1osceee

c

e vmnUn

nF

)( 2oscee vmnP

•The two forces are comparable when vosc~ve.

Collisional Absorption•e- oscillate at the laser frequency. Except high-power lasers, the E-field dominates.

tvv osc sin

•A simple model.

•Direction of e- randomized every e.disordering or thermalizing the energy of oscillation.

•Each e- contributes (1/2)mevosc2/ e.

•Thermal energy change per volume per second (1/2)menevosc2 / e.

Resonance Absorption•Dispersion relation for EM waves in a plasma.

2222 ckp

•The local value of p increases as the wave penetrates to higher density plasma until p = (the critical surface).

•EM wave resonantly excites a plasma wave.

•Plasma wave is damped to transfer energy.

•At densities greater than nc, k is imaginary. The wave decays evanescently.

•EM wave transverse. Plasma wave longitudinal.

•A laser normal to the critical surface cannot excite a plasma wave.

•Laser must enter obliquely with the correct polarization to give a component of the EM wave in the direction perpendicular to the critical surface.

•Plasma wave grows at the critical surface until a dampening process balances. Normally non-linear effect. Produces relatively few very hot e-. Problem for ICF. Long mean free path.

•Minimized by keeping I relatively low and short.

327

2

101.1

m

mne

Transport

Dendy, pg. 332.

•Laser energy absorbed at densities 0.0-0.1 times solid density. EM radiation cannot penetrate beyond the critical surface.

Laser-driven Instabilities•Brillouin instability.

•QM: Decay of a photon into a phonon and reflected photon.

•Classical: laser reflects off of the density fluctuation that is part of the ion-acoustic wave. The reflection beats with the laser beam to drive the ion acoustic wave.

•In a uniform density solid, e- are excited and reradiate dipole radiation. The dipole radiation recombines such that only the forward component is non-zero. The ion-acoustic wave causes longitudinal density variations. These variations create a non-zero backward component or reflection. The reflection beats with the laser and increases the density fluctuations via the ponderomotive force.

•Raman instability

•Photon decays into a plasmon (e- plasma wave quantum) and a photon.

•Plasma wave decays by producing high energy e-. Problematic for ICF.

•2 plasmon instability. Photon decays into 2 plasmons.

Rayleigh-Taylor Instability•Hydrodynamic instability critical to imploding targets.

•Similar to squeezing a balloon with your hands.

•Dense shell is accelerated inward by high pressure created in the ablating corona.

•At the interface, ripples develop and grow.

•Regions of high and low density fluid exchange position.

•Can lead to the shell breaking up prior to achieving complete compression.

Dendy, pg. 335.

•Simple model.

•High and low density fluid separated by a discontinuity.

•Perturbed sinusoidally with wave number k. Growth rate = (kg)1/2.

•Short wavelength grow fastest but merely smear the boundary.

•Most “dangerous” mode has with half-wavelength equal to R. = (g/R)1/2.

•Ti = (2R/g)1/2. Instability undergoes Ti = (2R/R)1/2 e-folds.

•Through experiment, the actual growth rate is 50% of the above.

•Implosion dynamics dictate R/R>20. The perturbation grows by exp(30)1/2 = 240.

•Suggests that the surface must be uniform down to at least 1% accuracy.

Rayleigh-Taylor Instability

Questions?