ϕ k,ω ¼
S k,ω
D k,ω
ð2:6:12Þ
D k,ω ¼ 1 þ
ω
2
pe
k
2
Z
1
ω À k Á u
k Á
∂g 0
∂u
du
where the denominator represents the dynamic screening effect by electron clouds. It
is noted that in the limit ω ! 0 , one can obtain the Debye screening given in (2.4.8).
Dawson and Oberman derived the absorption coefficient by calculating the joule
heating from the electron current derived by the equation of motion of electrons:
∂
∂t
j ¼
ω
2
pe
4π
E
Then, time average of Joule heating hj Á Ei gives the absorption rate of laser by
plasmas. Finally the energy absorption rate ν E relating to the collision frequency ν ei
is obtained in the form:
ν E ¼
ω
2
pe
ω 2
0
ν ei
ð2:6:13Þ
ν ei
ω pe
¼
Z
3
1
2π
3=2 1
nλ
3
De
ln Λ
ð2:6:14Þ
Λ ¼
ffiffi ffi
2
p
k max ω pe =k De ω 0
À
Á
In deriving (2.6.13), the following assumption has been used:
1. Higher harmonic components in (2.6.11) are neglected.
2. Bessel function is approximated J
2
1 k Á ε
ð
Þ % k Á ε
ð
Þ
2 by assuming that the
quivering distance is much shorter than the laser wavelength.
3. Laser intensity is assumed week enough to satisfy v os /v e < < 1.
It is informative to compare the collision frequency in (2.6.14) with one obtained
previously in (2.4.20):
ν ei 2:4:20
ð
Þ
ν ei 2:6:14
ð
Þ
¼
3
2
π
2
1=2 ¼ 1:89
ð2:6:15Þ
The relation (2.6.15) is except Coulomb log definition. It is noted that the evaluation
of lnΛ is also another discussion point.
2.6 Laser Absorption in Plasma
77
S k,ω
D k,ω
ð2:6:12Þ
D k,ω ¼ 1 þ
ω
2
pe
k
2
Z
1
ω À k Á u
k Á
∂g 0
∂u
du
where the denominator represents the dynamic screening effect by electron clouds. It
is noted that in the limit ω ! 0 , one can obtain the Debye screening given in (2.4.8).
Dawson and Oberman derived the absorption coefficient by calculating the joule
heating from the electron current derived by the equation of motion of electrons:
∂
∂t
j ¼
ω
2
pe
4π
E
Then, time average of Joule heating hj Á Ei gives the absorption rate of laser by
plasmas. Finally the energy absorption rate ν E relating to the collision frequency ν ei
is obtained in the form:
ν E ¼
ω
2
pe
ω 2
0
ν ei
ð2:6:13Þ
ν ei
ω pe
¼
Z
3
1
2π
3=2 1
nλ
3
De
ln Λ
ð2:6:14Þ
Λ ¼
ffiffi ffi
2
p
k max ω pe =k De ω 0
À
Á
In deriving (2.6.13), the following assumption has been used:
1. Higher harmonic components in (2.6.11) are neglected.
2. Bessel function is approximated J
2
1 k Á ε
ð
Þ % k Á ε
ð
Þ
2 by assuming that the
quivering distance is much shorter than the laser wavelength.
3. Laser intensity is assumed week enough to satisfy v os /v e < < 1.
It is informative to compare the collision frequency in (2.6.14) with one obtained
previously in (2.4.20):
ν ei 2:4:20
ð
Þ
ν ei 2:6:14
ð
Þ
¼
3
2
π
2
1=2 ¼ 1:89
ð2:6:15Þ
The relation (2.6.15) is except Coulomb log definition. It is noted that the evaluation
of lnΛ is also another discussion point.
2.6 Laser Absorption in Plasma
77