4.10.1 Effect of Inhomogeneity
Before solving (4.10.2), it’s intuitively understood that the density inhomogeneity
reduces the growth rate compared to that in the uniform plasma, since the decayed
plasma waves mainly localize near the matching region. Their spatial structures may
be determined by an eigen value problem like a localized wave function of
Schrodinger equation. In addition, the convective loss of the waves also reduces
the growth rate as additional loss process in the parametric instability growth. Note
that such additional loss process increases the threshold intensity of the instability.
This is the same image as saturation amplitude of the plasma wave in the linear mode
conversion as seen in Figs. 3.28 and 4.5(b), where the amplitude of the generated
plasma wave decreases as its group velocity increases.
Assume that the three-wave matching condition in Fig. 4.6 is satisfied for the
backward SRS with θ 1 ¼ θ 2 ¼ 0. Then, the absolute instability has the form:
γ 0 ¼
ω p0
4
ffiffiffiffiffiffiffiffiffiffi
ω 1 ω 2
p
kV os
ð4:10:4Þ
Taking a Laplace transformation in time of (4.10.2) with the Laplace transformation
variable p, neglecting the damping terms, and eliminating A 1 from (4.10.2), the
following equation is obtained:
∂
2
∂x 2 A 2 þ K
2 A 2 ¼ 0,
ð4:10:5Þ
where
K
2
¼
γ
2
0
V g1 V g2
À
p
2
4
1
V g1
þ
1
V g2
2
ð4:10:6Þ
For given p and the boundary condition that A 2 ¼ 0 at x ¼ 0 and L, (4.10.5) is easily
solved to have the solution:
A 2 ¼ sin
πn
L
x
,
πn
L
2 ¼ K
2 ,
ð4:10:7Þ
where n is an integer. It is clear that for p
2 > 0, the solution is unstable. Including the
damping term, instability condition is obtained:
γ
2
0
V g1 V g2
>
π
L
2 þ
1
4
ν 1
V g1
þ
ν 2
V g2
2
ð4:10:8Þ
It is clear that the matching condition in an inhomogeneous plasmas increases the
threshold intensity. In obtaining the growth rate, we have to solve (4.10.5) as an
4.10 Physics of Saturation of SRS Instability
159
Before solving (4.10.2), it’s intuitively understood that the density inhomogeneity
reduces the growth rate compared to that in the uniform plasma, since the decayed
plasma waves mainly localize near the matching region. Their spatial structures may
be determined by an eigen value problem like a localized wave function of
Schrodinger equation. In addition, the convective loss of the waves also reduces
the growth rate as additional loss process in the parametric instability growth. Note
that such additional loss process increases the threshold intensity of the instability.
This is the same image as saturation amplitude of the plasma wave in the linear mode
conversion as seen in Figs. 3.28 and 4.5(b), where the amplitude of the generated
plasma wave decreases as its group velocity increases.
Assume that the three-wave matching condition in Fig. 4.6 is satisfied for the
backward SRS with θ 1 ¼ θ 2 ¼ 0. Then, the absolute instability has the form:
γ 0 ¼
ω p0
4
ffiffiffiffiffiffiffiffiffiffi
ω 1 ω 2
p
kV os
ð4:10:4Þ
Taking a Laplace transformation in time of (4.10.2) with the Laplace transformation
variable p, neglecting the damping terms, and eliminating A 1 from (4.10.2), the
following equation is obtained:
∂
2
∂x 2 A 2 þ K
2 A 2 ¼ 0,
ð4:10:5Þ
where
K
2
¼
γ
2
0
V g1 V g2
À
p
2
4
1
V g1
þ
1
V g2
2
ð4:10:6Þ
For given p and the boundary condition that A 2 ¼ 0 at x ¼ 0 and L, (4.10.5) is easily
solved to have the solution:
A 2 ¼ sin
πn
L
x
,
πn
L
2 ¼ K
2 ,
ð4:10:7Þ
where n is an integer. It is clear that for p
2 > 0, the solution is unstable. Including the
damping term, instability condition is obtained:
γ
2
0
V g1 V g2
>
π
L
2 þ
1
4
ν 1
V g1
þ
ν 2
V g2
2
ð4:10:8Þ
It is clear that the matching condition in an inhomogeneous plasmas increases the
threshold intensity. In obtaining the growth rate, we have to solve (4.10.5) as an
4.10 Physics of Saturation of SRS Instability
159