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A similar equationholds for x2:d2 x2þ ω22 x2 ¼ c2 x1 E0dt2ð8:50Þ0Let x1 ¼ x1 cos ωt, x2 ¼ x2 cos ω t, and E0 ¼ E0 cos ω0 t. Equation (8.50) becomes002ω22 ω x2 cos ω t ¼ c2 E0 x1 cos ω0 t cos ωtð8:51Þ1¼ c2 E0 x1 f cos ½ðω0 þ ωÞt þ cos ½ðω0 ωÞtg2The driving terms on the right can excite oscillators x2 with frequencies0ω ¼ ω0 ωð8:52Þ8.5 Parametric Instabilities289In the absence of nonlinear interactions, x2 can only have the frequency ω2, so wemust have ω0 ¼ ω2. However, the driving terms can cause a frequency shift so thatω0 is only approximately equal to ω2.

Furthermore, ω0 can be complex, since there isdamping (which has been neglected so far for simplicity), or there can be growth(if there is an instability). In either case, x2 is an oscillator with finite Q and canrespond to a range of frequencies about ω2. If ω is small, one can see fromEq. (8.52) that both choices for ω0 may lie within the bandwidth of x2, and onemust allow for the existence of two oscillators, x2(ω0 + ω) and x2(ω0 ω).00Now let x1 ¼ x1 cos ω t and x2 ¼ x2 cos ½ðω0 ωÞt and insert into Eq. (8.49):00 200ω21 ω x1 cos ω t 1¼ c1 E0 x2 cos f½ω0 þ ðω0 ωÞtg þ cos ω0 ω0 ω t¼21c1 E0 x2 f cos ½ð2ω02ð8:53Þ ωÞt þ cos ωtgThe driving terms can excite not only the original oscillation x1(ω), but also newfrequencies ω00 ¼ 2ω0 ω.

We shall consider the case jω0j jω1j, so that 2ω0 ωlies outside the range of frequencies to which x1 can respond, and x1(2ω0 ω) canbe neglected. We therefore have three oscillators, x1(ω), x2(ω0 ω), and x2(ω0 + ω),which are coupled by Eqs. (8.49) and (8.50):ω21 ω2 x1 ðωÞ c1 E0 ðω0 Þ½x2 ðω0 ωÞ þ x2 ðω0 þ ωÞ ¼ 0hiω22 ðω0 ωÞ2 x2 ðω0 ωÞ c2 E0 ðω0 Þx1 ðωÞ ¼ 0hiω22 ðω0 þ ωÞ2 x2 ðω0 þ ωÞ c2 E0 ðω0 Þx1 ðωÞ ¼ 0ð8:54ÞThe dispersion relation is given by setting the determinant of the coefficients equalto zero: 2 ω ω21 c 2 E0 c E2 0c 1 E0ðω0 ωÞ2 ω220c 1 E0¼0022ðω0 þ ωÞ ω2ð8:55ÞA solution with Im(ω) > 0 would indicate an instability.For small frequency shifts and small damping or growth rates, we can set ω andω0 approximately equal to the undisturbed frequencies ω1 and ω2. Equation (8.52)then gives a frequency matching condition:ω0 ω 2 ω 1ð8:56ÞWhen the oscillators are waves in a plasma, ωt must be replaced by ωt k · r.

Thereis then also a wavelength matching condition2908 Nonlinear Effectsk0 k2 k1ð8:57Þdescribing spatial beats; that is, the periodicity of points of constructive anddestructive interference in space. The two conditions Eqs. (8.56) and (8.57) areeasily understood by analogy with quantum mechanics. Multiplying the former byPlanck’s constant h, we havehω0 ¼ hω2 hω1ð8:58ÞE0 and x2 may, for instance, be electromagnetic waves, so that hω0 and hω2 are thephoton energies. The oscillator x1 may be a Langmuir wave, or plasmon, withenergy hω1.

Equation (8.54) simply states the conservation of energy. Similarly,Eq. (8.53) states the conservation of momentum hk.For plasma waves, the simultaneous satisfaction of Eqs. (8.52) and (8.53) inone-dimensional problems is possible only for certain combinations of waves. Therequired relationships are best seen on an ω–k diagram (Fig. 8.14). Figure 8.14ashows the dispersion curves of an electron plasma wave ω2 (Bohm–Gross wave)Fig. 8.14 Parallelogram constructions showing the ω- and k-matching conditions for four parametric instabilities: (a) electron decay instability, (b) parametric decay instability, (c) stimulatedBrillouin backscattering instability, and (d) two-plasmon decay instability. In each case, ω0 is theincident wave, and ω1 and ω2 the decay waves.

The straight lines are the dispersion relation for ionwaves; the narrow parabolas, that for light waves; and the wide parabolas, that for electron waves8.5 Parametric Instabilities291and an ion acoustic wave ω1 (cf. Fig. 4.13). A large-amplitude electron wave (ω0,k0) can decay into a backward moving electron wave (ω2, k2) and an ion wave (ω1,k1). The parallelogram construction ensures that ω0 ¼ ω1 + ω2 and k0 ¼ k1 + k2. Thepositions of (ω0, k0) and (ω2, k2) on the electron curve must be adjusted so that thedifference vector lies on the ion curve. Note that an electron wave cannot decay intotwo other electron waves, because there is no way to make the difference vector lieon the electron curve.Figure 8.14b shows the parallelogram construction for the “parametric decay”instability.

Here, (ω0, k0) is an incident electromagnetic wave of large phasevelocity (ω0/k0 c). It excites an electron wave and an ion wave moving in oppositedirections. Since jk0j is small, we have jk1j jk2j and ω0 ¼ ω1 + ω2 for thisinstability.Figure 8.14c shows the ω–k diagram for the “parametric backscattering” instability, in which a light wave excites an ion wave and another light wave moving inthe opposite direction. This can also happen when the ion wave is replaced by aplasma wave. By analogy with similar phenomena in solid state physics, theseprocesses are called, respectively, “stimulated Brillouin scattering” and “stimulatedRaman scattering.”Figure 8.14d represents the two-plasmon decay instability of an electromagneticwave.

Note that the two decay waves are both electron plasma waves, so thatfrequency matching can occur only if ω0 ’ 2ωp. Expressed in terms of density,this condition is equivalent to n ’ nc/4, when nc is the critical density (Eq. (4.88))associated with ω0. This instability can therefore be expected to occur only near the“quarter-critical” layer of an inhomogeneous plasma.8.5.3Instability ThresholdParametric instabilities will occur at any amplitude if there is no damping, but inpractice even a small amount of either collisional or Landau damping will preventthe instability unless the pump wave is rather strong.

To calculate the threshold, onemust introduce the damping rates Γ1 and Γ2 of the oscillators x1 and x2. Equation(8.48) then becomesd 2 x1dx1¼0þ ω21 x1 þ 2Γ1dt2dtð8:59ÞFor instance, if x1 is the displacement of a spring damped by friction, the last termrepresents a force proportional to the velocity. If x1 is the electron density in aplasma wave damped by electron–neutral collisions, Γ1 is vc/2 (cf. Problem 4.5).Examination of Eqs. (8.49), (8.50), and (8.54) will show that it is all right to useexponential notation and let d/dt ! iω for x1 and x2, as long as we keep E0 real andallow x1 and x2 to be complex. Equations (8.49) and (8.50) become2928 Nonlinear Effectsω21 ω2 2iωΓ1 x1 ðωÞ ¼ c1 x2 E0hiω22 ðω ω0 Þ2 2iðω ω0 ÞΓ2 x2 ðω ω0 Þ ¼ c2 x1 E0ð8:60ÞWe further restrict ourselves to the simple case of two waves—that is, when ω ’ ω1and ω0 ω ’ ω2 but ω0 + ω is far enough from ω2 to be nonresonant—in whichcase the third row and column of Eq.

(8.55) can be ignored. If we now express x1,x2, and E0 in terms of their peak values, as in Eq. (8.53), a factor of 1/2 appears onthe right-hand sides of Eq. (8.60). Discarding the nonresonant terms and eliminating x1 and x2 from Eq. (8.60), we obtainih21ω2 ω21 þ 2iωΓ1 ðω0 ωÞ2 ω22 2iðω0 ωÞΓ2 ¼ c1 c2 E04ð8:61ÞAt threshold, we may set Im(ω) ¼ 0. The lowest threshold will occur at exactfrequency matching; i.e., ω ¼ ω1, ω0 ω ¼ ω2.

Then Eq. (8.61) gives 2c 1 c 2 E0thresh¼ 16ω1 ω2 Γ1 Γ2ð8:62ÞThe threshold goes to zero with the damping of either wave.Problems8.8 Prove that stimulated Raman scattering cannot occur at densities above nc/4.8.9 Stimulated Brillouin scattering is observed when a Nd-glass laser beam(λ ¼ 1.06 μm) irradiates a solid D2 target (Z ¼ 1, M ¼ 2MH). The backscatteredlight is red-shifted by 21.9 Å. From x-ray spectra, it is determined thatKTe ¼ 1 keV. Assuming that the scattering occurs in the region whereω2p << ω2 , and using Eq. (4.41) with γ i ¼ 3, make an estimate of the iontemperature.8.10 For stimulated Brillouin scattering (SBS), we may let x1 in Eq. (8.60) stand forthe ion wave density fluctuation n1, and x2 for the reflected wave electric fieldE2. The coupling coefficients are then given byc1 ¼ ε0 k21 ω2p =ω0 ω2 Mc2 ¼ ω2p ω2 =n0 ω0and threshold pump intensity in a homogeneous plasma is given by Eq.

(8.62).This is commonly expressed in terms of hv2osc i, the rms electron oscillationvelocity caused by the pump wave (cf. Eq. (8.37)):vosc eE0 =mω0The damping rate Γ2 can be found from Problem (4.37b) for v/ω 1.8.5 Parametric Instabilities293(a) Show that, for Ti Te and v2e KT e =m, the SBS threshold is given byv2osc4Γ1 ν¼ω1 ω2v2ewhere ω1 ¼ k1vs and Γ1 is the ion Landau damping rate given byEq. (7.133).(b) Calculate the threshold laser intensity I0 in W/cm2 for SBS of CO2(10.6 μm) light in a uniform hydrogen plasma with Te ¼ 100 eV,Ti ¼ 10 eV, and n0 ¼ 1023 m3 (Hint: Use the Spitzer resistivity toevaluate vei.)8.11 The growth rate of stimulated Brillouin scattering in a homogeneous plasmafar above threshold can be computed from Eq. (8.61) by neglecting thedamping terms.

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