δ m ¼ 1 À
1
γ
h i
ð6:5:10Þ
For weak intensity limit, Taylor expanding (6.5.8) and (6.5.10) to the amplitude a 0 ,
the divergence term and focusing terms by ponderomotive force and relativistic mass
correction effect are given in the form:
δ r : δ n : δ m ¼
ρ
2
s
R
2
:
ρ
2
s
R
2
a
2
0
4
:
a
2
0
4
ð6:5:11Þ
It is clear that in the low-intensity limit, the refraction is dominant, and the laser
beam diverges, while the relativistic and ponderomotive term becomes dominant to
focus the laser beam due to nonlinear change of the refractive index. This is called
self-focusing phenomenon.
It is important to pay attention to the radius dependence of three contribution in
(6.5.11). In the low density and a 0 < 1, the diffraction term dominates than the two
focusing terms, and no self-focusing may not happen. However, as the increase of
density and laser strength, the focusing terms become dominant to overwhelm the
diffraction term. The physics of laser propagation appears to be complicated in
propagating in the high-density with relativistic intensity.
It is important to point out that there is a critical laser power for self-focusing
condition, P cr in the limit of weak laser intensity (a 0 << 1). When the relativistic
effect balances with the diffraction effect, the following condition should be
satisfied:
a
2
0 R
2
¼ 4ρ
2
s , P cr / ω
2 a
2
0 R
2
/
n cr
n 0
ð6:5:12Þ
More precise calculation yields this critical power of cylindrical laser beams for selffocusing condition in the form:
P cr ¼ 17:5
n cr
n 0
GW
½
ð6:5:13Þ
This critical power is relatively low compared to the present-day power of ultrashort ultra-intense lasers, 100TW ~10PW. If such laser is irradiated on the plasmas,
we have to note that the self-focusing always is taken place in the plasma.
6.5.2 Strong Laser Limit
The above evaluation is based on (6.5.4), where we have assumed that a 0 << 1. Let
us consider the opposite case for a 0 >> 1. In this limit, the ratio shown in (6.5.11)
changes as
224
6 Relativistic Laser Plasma Interactions
1
γ
h i
ð6:5:10Þ
For weak intensity limit, Taylor expanding (6.5.8) and (6.5.10) to the amplitude a 0 ,
the divergence term and focusing terms by ponderomotive force and relativistic mass
correction effect are given in the form:
δ r : δ n : δ m ¼
ρ
2
s
R
2
:
ρ
2
s
R
2
a
2
0
4
:
a
2
0
4
ð6:5:11Þ
It is clear that in the low-intensity limit, the refraction is dominant, and the laser
beam diverges, while the relativistic and ponderomotive term becomes dominant to
focus the laser beam due to nonlinear change of the refractive index. This is called
self-focusing phenomenon.
It is important to pay attention to the radius dependence of three contribution in
(6.5.11). In the low density and a 0 < 1, the diffraction term dominates than the two
focusing terms, and no self-focusing may not happen. However, as the increase of
density and laser strength, the focusing terms become dominant to overwhelm the
diffraction term. The physics of laser propagation appears to be complicated in
propagating in the high-density with relativistic intensity.
It is important to point out that there is a critical laser power for self-focusing
condition, P cr in the limit of weak laser intensity (a 0 << 1). When the relativistic
effect balances with the diffraction effect, the following condition should be
satisfied:
a
2
0 R
2
¼ 4ρ
2
s , P cr / ω
2 a
2
0 R
2
/
n cr
n 0
ð6:5:12Þ
More precise calculation yields this critical power of cylindrical laser beams for selffocusing condition in the form:
P cr ¼ 17:5
n cr
n 0
GW
½
ð6:5:13Þ
This critical power is relatively low compared to the present-day power of ultrashort ultra-intense lasers, 100TW ~10PW. If such laser is irradiated on the plasmas,
we have to note that the self-focusing always is taken place in the plasma.
6.5.2 Strong Laser Limit
The above evaluation is based on (6.5.4), where we have assumed that a 0 << 1. Let
us consider the opposite case for a 0 >> 1. In this limit, the ratio shown in (6.5.11)
changes as
224
6 Relativistic Laser Plasma Interactions