458 Chapter 15. Wave-wave nonlinearities
discussed in this problem can be expressed in terms of Jacobi elliptic functions
(Sagdeev and Galeev 1969).- Stimulated Raman scattering in laser fusion: Laser fusion is a proposed method for
attaining a controlled fusion reaction. This method involves illuminating a millimeter
radius pellet with intense laser light so as to ablate the outer layer of the pellet. The
radial outflow of the ablating material constitutes a radial outflow of momentum and
so, in order to conserve radial momentum, the rest of the pellet accelerates inwards
in rocket-like fashion. An important issue is the decay of the incident laser light
into other waves because such a decay would reduce the power available todrive
the ablation process. For example there could be stimulated Raman scattering where
the incident laser light decays into a outward propagating electromagneticwave (the
backscattered light) and an inward propagating Langmuir wave.
(a) Let the incident laser beam be denoted by subscript 3, the outward propagat-
ing electromagnetic wave be denoted by subscript 2 and the inward propagating
Langmuir wave be denoted by subscript 1. Assume that the laser wavelength is
much shorter than the characteristic density scale length. Show that if the back-
ward scattered electromagnetic wave has a frequency only slightly aboveωpe
then it is possible to satisfy the frequency and wavenumber matching conditions
at a location where the density is approximately 1/4 of the density where the
incident wave would reflect.
(b) Draw a sketch ofωversuskfor the incident em wave, the backscattered em
wave, and the Langmuir wave. Note thatvTe<<cso that if this plot is scaled to
show the dispersion of the electromagnetic waves, the Langmuir wave dispersion
is almost a horizontal line.
(c) Draw vectors on the sketch in (a) with coordinates{k,ω}so that the incident
electromagnetic wave is a vectorv 3 ={k 3 ,ω 3 },the backscattered wave is a
vectorv 2 ={k 2 ,ω 2 }and the Langmuir wave is a vectorv 1 ={k 1 ,ω 1 }.Show
on the sketch how the vectors can add up in a manner consistent with the selec-
tion rulesv 3 =v 1 +v 2. - Parametric decay instability: The minimum pump amplitude for the parametric decay
instability (electromagnetic wave decays into a Langmuir wave and anion acoustic
wave) is given by Eq.(15.119) to be
E 3 =4
√
ω 1 ω 2 Γ 1 Γ 2
λ
where the coupling parameter was defined in Eq.(15.81) to beλ=
ωpeqk 1
√
mimeω 2 ω 3.
(a) How do the Langmuir wave frequencyω 2 and the ion acoustic wave frequency
ω 1 compare toωpe(nearly same, much larger, or much smaller)?
(b) Taking into account the selection rules, how does the pump frequencyω 3 com-
pare toωpe?What does this imply fork 3 and hencek 1 andk 2?