6.7 JxB Force and Heating
When a linearly polarized laser with relativistic intensity is irradiated on a solid
surface, it is well-known that the v 3 B force works to the direction of laser
propagation. Since the velocity and magnetic field oscillate with the laser frequency
ω, the ponderomotive force shown in (6.3.4) gives the momentum to the electrons
near the surface. In addition, it is clear that the force oscillating with 2ω also works to
the electrons. The latter force is called JxB force [5].
This force is calculated in general as follows:
f ¼ Àev  B
v ¼
p ⊥
mγ
¼ À
e
mγ
A
ð6:7:1Þ
Therefore, the force f reduces to
f ¼ Àmc
2
∇
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
a 2 þ 1
p
,
f ¼ Àmc
2
∇γ
ð6:7:2Þ
Compare (6.7.2) to (6.3.10). In weak relativistic case (a
2 < <1), the ponderomotive
and the force f are clearly derived from (6.7.2) in the form:
f ¼ À
mc
2
4
∇ a 0 r
ð Þ
j
j
2 1 þ cos 2ωt
ð Þ
½
ð 6:7:3Þ
The first term in RHS of (6.7.3) is the ponderomotive force defined in (6.3.5), while
the second term is JxB force. Inserting sinusoidal oscillation of a in (6.7.2) for
arbitrary amplitude, the relativistic nonlinear forces are given:
f ¼ Àmc
2
∇ 1 þ
1
2
a 0 r
ð Þ
j
j
2 1 þ cos 2ωt
ð Þ
ð
Þ
h
i 1=2
ð6:7:4Þ
Note that in the relativistic case, not only 2ω but also 4ω and other higher harmonic
forces appear. In the limit of highly relativistic intensity, (6.7.4) is approximated to
f ¼ À
mc
2
ffiffi ffi
2
p cos ωt
ð Þ
j
j ∇ a 0 r
ð Þ
j
j
ð6:7:5Þ
It is obvious that when the laser is irradiated on the solid surface, the
ponderomotive force keeps pushing the electrons toward the inside of the solid.
In the limit of ultra-short pulse, we can assume that the ions are immobile, and the
charge separation creates the ambipolar electrostatic field as mentioned in Sect. 6.1.
Including the pull-back force by the electrostatic field, the equation of the electron at
the solid surface is approximately given from (6.1.10) in the form in the
non-relativistic limit:
6.7 JxB Force and Heating
231
When a linearly polarized laser with relativistic intensity is irradiated on a solid
surface, it is well-known that the v 3 B force works to the direction of laser
propagation. Since the velocity and magnetic field oscillate with the laser frequency
ω, the ponderomotive force shown in (6.3.4) gives the momentum to the electrons
near the surface. In addition, it is clear that the force oscillating with 2ω also works to
the electrons. The latter force is called JxB force [5].
This force is calculated in general as follows:
f ¼ Àev  B
v ¼
p ⊥
mγ
¼ À
e
mγ
A
ð6:7:1Þ
Therefore, the force f reduces to
f ¼ Àmc
2
∇
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
a 2 þ 1
p
,
f ¼ Àmc
2
∇γ
ð6:7:2Þ
Compare (6.7.2) to (6.3.10). In weak relativistic case (a
2 < <1), the ponderomotive
and the force f are clearly derived from (6.7.2) in the form:
f ¼ À
mc
2
4
∇ a 0 r
ð Þ
j
j
2 1 þ cos 2ωt
ð Þ
½
ð 6:7:3Þ
The first term in RHS of (6.7.3) is the ponderomotive force defined in (6.3.5), while
the second term is JxB force. Inserting sinusoidal oscillation of a in (6.7.2) for
arbitrary amplitude, the relativistic nonlinear forces are given:
f ¼ Àmc
2
∇ 1 þ
1
2
a 0 r
ð Þ
j
j
2 1 þ cos 2ωt
ð Þ
ð
Þ
h
i 1=2
ð6:7:4Þ
Note that in the relativistic case, not only 2ω but also 4ω and other higher harmonic
forces appear. In the limit of highly relativistic intensity, (6.7.4) is approximated to
f ¼ À
mc
2
ffiffi ffi
2
p cos ωt
ð Þ
j
j ∇ a 0 r
ð Þ
j
j
ð6:7:5Þ
It is obvious that when the laser is irradiated on the solid surface, the
ponderomotive force keeps pushing the electrons toward the inside of the solid.
In the limit of ultra-short pulse, we can assume that the ions are immobile, and the
charge separation creates the ambipolar electrostatic field as mentioned in Sect. 6.1.
Including the pull-back force by the electrostatic field, the equation of the electron at
the solid surface is approximately given from (6.1.10) in the form in the
non-relativistic limit:
6.7 JxB Force and Heating
231