The change of physical values over the region (8.3.7) is shown in red in Fig. 8.6.
During this short time, the dephasing rate R is drastically reduced as seen in
Fig. 8.6c. Since the amplitude “a” is proportional to cos(ξ) in (8.3.5), it is clear
from Fig. 8.6 that electron acceleration after the interaction (x > 150 mm) is taken
place almost in the same phase of the laser:
ξ t, x
ð Þ ¼ ω t À x=c
ð
Þ¼ b t À b x % const:
ð8:3:11Þ
This is nothing without the laser direct acceleration (LDA), where electron velocity is nearly equal to the speed of light, and it can continue to increase its energy from
(8.1.3) as seen in Fig. 8.6d after the red line timing. In the case of Fig. 8.6, the
condition (Àv y E y > 0) is maintained over the traveling long distance and traveling
time t ¼ x/c. How much energy the electron obtains due to such process depends on
the size of plasma or particle acceleration distance. The calculation is done up to
1000 μm in Fig. 8.6, which is about 0.3 GeV from Fig. 8.6d and not long enough to
be accelerated to the maximum energy theoretically evaluated in (8.3.10).
It is noted that in Fig. 7.26, we have already observed spiky electrostatic field
generation in the under-dense plasma produced from the foam material. This is a
good example how the electrostatic field in the direction of laser propagation is
produced. Of course, the electric field is not only in the favor direction for acceleration to the laser propagation direction. However, some fraction of electrons can be
accelerated locally by the electrostatic field and continue to be accelerated by the
laser direct acceleration. Such mechanism is also physical candidate to explain the
production of super-ponderomotive hot electrons.
When a longitudinal electric field (the electric field in the x-direction) is located in
the laser-plasma interaction region, it is suggested that the hot electron energy is
higher energy than without the longitudinal field. This is happen dominantly near the
timing, p x ¼ p y ¼ 0 in (8.2.2) and (8.2.3). The addition of the longitudinal field E x in
(8.1.7) localized over the interval Δx changed the momentum in a short time like
Δp x ¼
Z tþΔt
t
E x dt ¼
Z
xþΔx
x
E x
γ
p x
dx
ð8:3:12Þ
It is clear that the x-momentum change is biggest for the condition that
E x < 0
p x % 0
ð8:3:13Þ
300
8 Chaos due to Relativistic Effect
During this short time, the dephasing rate R is drastically reduced as seen in
Fig. 8.6c. Since the amplitude “a” is proportional to cos(ξ) in (8.3.5), it is clear
from Fig. 8.6 that electron acceleration after the interaction (x > 150 mm) is taken
place almost in the same phase of the laser:
ξ t, x
ð Þ ¼ ω t À x=c
ð
Þ¼ b t À b x % const:
ð8:3:11Þ
This is nothing without the laser direct acceleration (LDA), where electron velocity is nearly equal to the speed of light, and it can continue to increase its energy from
(8.1.3) as seen in Fig. 8.6d after the red line timing. In the case of Fig. 8.6, the
condition (Àv y E y > 0) is maintained over the traveling long distance and traveling
time t ¼ x/c. How much energy the electron obtains due to such process depends on
the size of plasma or particle acceleration distance. The calculation is done up to
1000 μm in Fig. 8.6, which is about 0.3 GeV from Fig. 8.6d and not long enough to
be accelerated to the maximum energy theoretically evaluated in (8.3.10).
It is noted that in Fig. 7.26, we have already observed spiky electrostatic field
generation in the under-dense plasma produced from the foam material. This is a
good example how the electrostatic field in the direction of laser propagation is
produced. Of course, the electric field is not only in the favor direction for acceleration to the laser propagation direction. However, some fraction of electrons can be
accelerated locally by the electrostatic field and continue to be accelerated by the
laser direct acceleration. Such mechanism is also physical candidate to explain the
production of super-ponderomotive hot electrons.
When a longitudinal electric field (the electric field in the x-direction) is located in
the laser-plasma interaction region, it is suggested that the hot electron energy is
higher energy than without the longitudinal field. This is happen dominantly near the
timing, p x ¼ p y ¼ 0 in (8.2.2) and (8.2.3). The addition of the longitudinal field E x in
(8.1.7) localized over the interval Δx changed the momentum in a short time like
Δp x ¼
Z tþΔt
t
E x dt ¼
Z
xþΔx
x
E x
γ
p x
dx
ð8:3:12Þ
It is clear that the x-momentum change is biggest for the condition that
E x < 0
p x % 0
ð8:3:13Þ
300
8 Chaos due to Relativistic Effect