8.2 Laser Direct Acceleration
When an electron is oscillating in a relativistic laser field, it can have a possibility to
be accelerated after an abrupt change of a constant of motion by an external force.
Since the electron is possibly accelerated by the laser field with new constant of
motions, it can be called a kind of laser direct acceleration. In Sect. 5.3, we have
assumed an adiabatic motion of free electron initially at rest before the laser filed is
applied. As shown in Appendix 2, the adiabatic relation is satisfied for
α ¼ 1
β ¼ 0
ð8:2:1Þ
Then, such adiabatic electron has the solution from (8.1.19) to (8.1.21) in the form:
p y ¼ a
ð8:2:2Þ
p x ¼
1
2
p
2
y ¼
1
2
a
2
ð8:2:3Þ
γ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p 2
x þ p 2
y þ 1
q
¼
1
2
a
2
þ 1
ð8:2:4Þ
Clearly the motion is periodic, and p x oscillates with 2ω frequency from a in (8.1.13).
The trajectory of the two-dimensional (p x , p y ) plane is parabolic as shown by (a) in
Fig. 8.1. The adiabatic motion of an electron in relativistic laser field has an average
energy and average drift velocity in the x-direction of the forms:
γ
h i À 1 ¼
1
4
a
2
0 ,
v x
h i ¼
a
2
0
a 2
0 þ 4
ð8:2:5Þ
It is noted that if the electron is ejected from the laser interaction region
non-adiabatically, it has the ejection angle θ from the x-direction as derived from
(8.2.2) to (8.2.4):
tan
2
θ ¼
p y
p x
2
¼
2
γ À 2
ð8:2:6Þ
This indicates that for a highly relativistic case, the accelerated electrons are ejected
with a narrow angle spread. In addition, the electron energy in (8.2.4) is about a 0
times higher than the ponderomotive scaling energy in (6.3.6).
292
8 Chaos due to Relativistic Effect
When an electron is oscillating in a relativistic laser field, it can have a possibility to
be accelerated after an abrupt change of a constant of motion by an external force.
Since the electron is possibly accelerated by the laser field with new constant of
motions, it can be called a kind of laser direct acceleration. In Sect. 5.3, we have
assumed an adiabatic motion of free electron initially at rest before the laser filed is
applied. As shown in Appendix 2, the adiabatic relation is satisfied for
α ¼ 1
β ¼ 0
ð8:2:1Þ
Then, such adiabatic electron has the solution from (8.1.19) to (8.1.21) in the form:
p y ¼ a
ð8:2:2Þ
p x ¼
1
2
p
2
y ¼
1
2
a
2
ð8:2:3Þ
γ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p 2
x þ p 2
y þ 1
q
¼
1
2
a
2
þ 1
ð8:2:4Þ
Clearly the motion is periodic, and p x oscillates with 2ω frequency from a in (8.1.13).
The trajectory of the two-dimensional (p x , p y ) plane is parabolic as shown by (a) in
Fig. 8.1. The adiabatic motion of an electron in relativistic laser field has an average
energy and average drift velocity in the x-direction of the forms:
γ
h i À 1 ¼
1
4
a
2
0 ,
v x
h i ¼
a
2
0
a 2
0 þ 4
ð8:2:5Þ
It is noted that if the electron is ejected from the laser interaction region
non-adiabatically, it has the ejection angle θ from the x-direction as derived from
(8.2.2) to (8.2.4):
tan
2
θ ¼
p y
p x
2
¼
2
γ À 2
ð8:2:6Þ
This indicates that for a highly relativistic case, the accelerated electrons are ejected
with a narrow angle spread. In addition, the electron energy in (8.2.4) is about a 0
times higher than the ponderomotive scaling energy in (6.3.6).
292
8 Chaos due to Relativistic Effect