V 0 ¼
eE 0
mω 0
,
V 1 ¼
eE 1
mω 1
ð4:6:7Þ
Taking time derivative of (3.5.6) and inserting the new velocity perturbation of
(4.6.6), the following equation to the plasma waves is obtained:
∂
2
∂t 2 À 3v
2
e
∂
2
∂x 2
2
þ ω
2
p0
δn 2
n 0
¼ Àcosθ 1 k
2
2 V 0 V 1
ð4:6:8Þ
And for the case of the ion waves, it is easy to obtain the equation:
∂
2
∂t 2 À C
2
s
∂
2
∂x 2
2
δn 2
n 0
¼ Àcosθ 1
m e
m i
k
2
2 V 0 V 1 ,
ð4:6:9Þ
where the third coordinate x 2 is introduced for the electrostatic waves to propagate in
this direction. It is noted that three coordinates have the geometrical relation in
Fig. 4.6, and the angle θ 1 is defined there.
It is useful to grasp the image of the parametric instability with the work by the
ponderomotive force in (4.6.6). Graphic depiction of a parametric instability such as
SRS or SBS is given in Fig. 4.7 [6]. Assume that θ 1 ¼ π and the laser is incident from
the right in (4.6.8) or (4.6.9). In step 1, the laser propagates from the right to left in a
Fig. 4.7 Graphic depiction
of parametric instabilities.
[6]
148
4 Nonlinear Physics of Laser-Plasma Interaction