442 Chapter 15. Wave-wave nonlinearities
High frequency wave is an electromagnetic wave Here the wave is transverse so
∇·E ̃=0and the curl of Faraday’s law becomes
∇^2 E ̃=
∂∇×B ̃
∂t. (15.70)
Substituting Eq.(15.41) and using the linear equation of motion gives the general expression
for a high frequency electromagnetic wave with non-linear coupling term,
∂^2 E ̃
∂t^2+ω^2 peE ̃−c^2 ∇^2 E=−1
ε 0∂
∂t(
̃nleqe ̃uhe)
; (15.71)
the term ̃nheqe ̃ulehas been dropped for the reasons given in the previous paragraph.
Summary of mode interactions Examination of the various combinations consid-
ered above shows that the non-linear coupling acting on high frequency modes (either pump
and high frequency daughter) is an effective modulation of the density experienced by the
high frequency wave, i.e.,ω^2 pe→ω^2 pe+ω^2 pen ̃l/n. On the other hand the non-linear cou-
pling acting on the low frequency daughter comes from the electron ponderomotive force
which effectively modulates the electron temperature experienced bythe low frequency
mode, i.e.,nκT ̃ e→ ̃nκTe+nκ
(
u ̃he) 2
/ 2.
Coupled oscillator formulation To see how the coupled wave equations can be ex-
pressed in terms of coupled oscillators, the specific situation of an electromagnetic wave
interacting with a Langmuir wave and an ion acoustic wave is now considered. In this case,
the three coupled equations are
∂^2 E ̃ 3
∂t^2+ω^2 peE ̃ 3 − c^2 ∇^2 E ̃ 3 = −qe
ε 0∂
∂t( ̃n 1 ̃u 2 )∂^2 E ̃ 2
∂t^2
+ω^2 peE ̃ 2 −3 κTe
me∇^2 E ̃ 2 = −
qe
ε 0∂
∂t
( ̃n 1 ̃u 3 )∂^2 ̃n 1
∂t^2−c^2 s∇^2 ̃n 1 = nme
mi∇^2 ( ̃u 2 · ̃u 3 ) (15.72)where the relation
(
1
2
(
u ̃he) 2
)
ω 1 ,k 1=
1
2
〈( ̃u 2 + ̃u 3 )·( ̃u 2 + ̃u 3 )〉=〈 ̃u 2 · ̃u 3 〉 (15.73)has been used and the angle brackets refer to the component oscillating at thebeat fre-
quencyω 1 =ω 3 −ω 2 and having the beat wavevectork 1 =k 3 −k 2.
The subscriptehas been dropped from all dependent variables because they all refer
to electrons. Since the acoustic wave frequency is much smaller thanωpe,it is possible to
approximate
∂
∂t
( ̃n 1 ̃u 2 )≃n ̃ 1∂ ̃u 2
∂t= ̃n 1qe
meE ̃ 2. (15.74)
Using the quiver relation Eq.(15.36), the product of the high frequency velocities can be
expressed as
〈 ̃u 2 · ̃u 3 〉=q^2 e
m^2 eω 2 ω 3