4.6.2 Thermal Noise of Electrostatic Fluctuations
It is known that any thermal system has noise of physical quantities. It is derived, for
example, in Ref. [7] that the fluctuation level of the electric field in one component
plasma in thermodynamically equilibrium state is given with the dynamic form
factor S(k,ω) defined for time correlation of the density fluctuation Fourier
components, namely,
S k, ω
ð
Þ ¼
1
2π
Z
dt δn k t
0
þ t
ð
Þδn Àk t
0
ð Þ
h
i exp iωt
ð Þ
ð4:6:11Þ
It is given in Eq. (9.29) of Ref. [7] in the form:
S k, ω
ð
Þ ¼
n e
πω
k
2
k
2
De
Im
1
ε k, ω
ð
Þ
&
'
,
ð4:6:12Þ
where ε is the dielectric constant. For a complex function ε ¼ ε r + iε i ,
Im
1
ε k, ω
ð
Þ
&
'
¼
ε i
ε 2
r þ ε 2
i
ð4:6:13Þ
The imaginary part in (4.6.12) has maximum for the fluctuation satisfying the
dispersion relation, ε r (k, ω) ¼ 0.
2.0
|A 0 |
|A 2 |
|A 1 |
1.5
1.0
0.5
2
4
time
6
8
Fig. 4.8 Schematics of time
evolution of the envelope of
three oscillators interacting
with a nonlinear coupling
150
4 Nonlinear Physics of Laser-Plasma Interaction