g t, ρ, u
ð
Þ¼g 0 þ g 1
It is reasonable to assume that g 0 is given with Maxwell distribution for the
velocity u.
Vlasov equation to the linear perturbation g 1 is:
∂g 1
∂t
þ u Á
∂g 1
∂ρ
þ
e
m
∂ϕ
∂ρ
Á
∂g 0
∂u
¼ 0
ð2:6:8aÞ
and Poisson equation becomes:
∂
2 ϕ
∂ρ 2 ¼ 4πe n 0
Z
g 1 d
3 u À Z
X
i
δ ρ À ε sin ω 0 t À r i
ð
Þ
(
)
ð2:6:8bÞ
Carrying out Fourier-Laplace transformation of (2.6.8a) and (2.6.8b), they become:
g k,ω ¼
e=m
ω À k Á u
k Á
∂g 0
∂u
ϕ k,ω
ð2:6:9Þ
ϕ k,ω ¼ À
n 0 e
ε 0 k
2
Z
g k,ω du þ S k,ω
ð2:6:10Þ
where S k, ω stems from the electric field by randomly located ions. It is calculated to
be:
S k,ω ¼ À
Ze
8π 3 ε 0 k
2
X 1
n¼À1
À1
ð Þ
n J n k Á ε
ð
Þ
ω À nω 0
X
j
exp Àik Á r j
À
Á
ð2:6:11Þ
In obtaining (2.6.11), Fourier transform is done to obtain at first:
Z
dρδ ρ À ε sin ω 0 t À r i
ð
Þ e
ikÁρ
¼ 8π
ð Þ
À1 e
ikÁ ε sin ω 0 tÀr i
ð
Þ
Then, the following mathematical relation is used:
e
iz sin ωt
X 1
n¼À1
À1
ð Þ
n J n z
ð Þe
inωt
It is noted that (2.6.11) has many poles at ω ¼ nω 0 which means that there are higher
harmonic oscillation components in the electric field.
Substituting (2.6.9) to (2.6.10) leads
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2 Laser Absorption by Coulomb Collision