magnetic field. In addition, Lorentz factor is constant; therefore, (5.3.23) is easily
solved to obtain the current density:
j
ε 0
¼ À
ω
2
p0
γ 0
A,
ð6:2:1Þ
where γ 0 is defined in (5.3.16). Inserting this current to (5.2.8), we obtain the linear
dispersion relation for electromagnetic waves in plasmas:
ω
2
¼ c
2 k
2
þ
ω
2
p0
γ 0
ð6:2:2Þ
Equation (6.2.2) shows that the effective plasma frequency becomes lower due to the
relativistic mass correction by which the electron becomes heavier in the relativistic
regime. This fact gives an important change of physics in laser-plasma interaction in
relativistic case. The cutoff density n cr determined only by the laser frequency
becomes also a function of Lorentz factor.
It is clear that the cutoff density effectively increases from n cr in non-relativistic
case to the form:
n
R
cr ¼ γ 0 n cr
ð6:2:3Þ
When the intensity is high enough, the cutoff density possibly becomes higher than
the plasma density of solid materials. Then, laser can penetrate though the solids, and
they become transparent to the laser light. This phenomena is relativistic transparency. When the matter becomes transparent to strong laser field, the electrons start to
drift as shown in Fig. 6.3. Then, for example, strong electrostatic field is produced on
the rear surface of matter, and this electrostatic field is used for ion acceleration by
relativistic lasers. In the ultra-relativistic limit, the contribution by the plasma or
matter to the dispersion relation of (6.2.2) disappears, and the laser propagates as if
the matters are transparent.
It is informative to estimate the critical laser intensity at which solid matters
become transparent to the laser with 1 μm laser wavelength (n cr ¼ 10
21 cm
À3 ). From
(6.2.3), we obtain the threshold value of a 0 so that solid material becomes transparent
in the form:
a 0 %
ffiffi ffi
2
p
n solid
10
21 cm À3
ð6:2:4Þ
The solid aluminum with the density 2.7 g/cm
3 has the electron number density
8 Â 10
23 cm
À3 . The corresponding critical laser intensity for the relativistic transparency is 10
24 W/cm
2 (a 0 ¼ 10
3 ). The liquid hydrogen has the density 0.07 g/cm
3 ,
and it becomes transparent at 2 Â 10
21 W/cm
2 . Plastic target with the solid density of
1 g/cm
3 has the critical intensity of 10
23 W/cm
2 . It is surprising that even in solid
208
6 Relativistic Laser Plasma Interactions
solved to obtain the current density:
j
ε 0
¼ À
ω
2
p0
γ 0
A,
ð6:2:1Þ
where γ 0 is defined in (5.3.16). Inserting this current to (5.2.8), we obtain the linear
dispersion relation for electromagnetic waves in plasmas:
ω
2
¼ c
2 k
2
þ
ω
2
p0
γ 0
ð6:2:2Þ
Equation (6.2.2) shows that the effective plasma frequency becomes lower due to the
relativistic mass correction by which the electron becomes heavier in the relativistic
regime. This fact gives an important change of physics in laser-plasma interaction in
relativistic case. The cutoff density n cr determined only by the laser frequency
becomes also a function of Lorentz factor.
It is clear that the cutoff density effectively increases from n cr in non-relativistic
case to the form:
n
R
cr ¼ γ 0 n cr
ð6:2:3Þ
When the intensity is high enough, the cutoff density possibly becomes higher than
the plasma density of solid materials. Then, laser can penetrate though the solids, and
they become transparent to the laser light. This phenomena is relativistic transparency. When the matter becomes transparent to strong laser field, the electrons start to
drift as shown in Fig. 6.3. Then, for example, strong electrostatic field is produced on
the rear surface of matter, and this electrostatic field is used for ion acceleration by
relativistic lasers. In the ultra-relativistic limit, the contribution by the plasma or
matter to the dispersion relation of (6.2.2) disappears, and the laser propagates as if
the matters are transparent.
It is informative to estimate the critical laser intensity at which solid matters
become transparent to the laser with 1 μm laser wavelength (n cr ¼ 10
21 cm
À3 ). From
(6.2.3), we obtain the threshold value of a 0 so that solid material becomes transparent
in the form:
a 0 %
ffiffi ffi
2
p
n solid
10
21 cm À3
ð6:2:4Þ
The solid aluminum with the density 2.7 g/cm
3 has the electron number density
8 Â 10
23 cm
À3 . The corresponding critical laser intensity for the relativistic transparency is 10
24 W/cm
2 (a 0 ¼ 10
3 ). The liquid hydrogen has the density 0.07 g/cm
3 ,
and it becomes transparent at 2 Â 10
21 W/cm
2 . Plastic target with the solid density of
1 g/cm
3 has the critical intensity of 10
23 W/cm
2 . It is surprising that even in solid
208
6 Relativistic Laser Plasma Interactions