p
max
x
%
1
2RÃ
a
2
0 þ 1
À
Á
ð8:3:2Þ
Setting that the y-momentum after just after passing the finite E x region is p y ¼ p y *,
the following relation should be satisfied just at this time:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p Ã
x
2 þ p Ã
y
2 þ 1
q
À p
Ã
x ¼ RÃ
ð 8:3:3Þ
Assuming p x * is much larger than unity, R* has the minimum value for the case with
p y * ¼ 0 in the approximate form:
RÃ %
1
2p Ã
x
ð8:3:4Þ
Note that the relation (8.3.4) is the same as (8.2.14) and the role of the longitudinal
electric field is to jump the initial condition to realize (8.2.14) by the acceleration of
electrons in the x-direction with E x (<0).
In order to see the detail dynamics described above, the basic Eqs. (8.1.1) and
(8.1.2) are numerically integrated [1]. The laser amplitude is modeled as a Gaussian
pulse in the form:
a x, t
ð Þ ¼ a 0 cosξexp À
ξ
2
2 ωτ L
ð
Þ
2
"
#
ð8:3:5Þ
The laser parameters are
a 0 ¼ 10, τ L ¼ 40fs, λ L ¼ 1μm
ð8:3:6Þ
In the computation a constant longitudinal electric field is assumed for an interval of
5 μm:
b
E x ¼ À0:1a 0 142 < x μm
ð Þ < 147
ð8:3:7Þ
The potential energy difference over this electric field is quite large to change the
dephasing rate R*:
Δγ ¼ 16MeV
ð8:3:8Þ
This is about the change of p x ¼ 31, namely, the maximum energy (8.2.15) is
roughly
298
8 Chaos due to Relativistic Effect
max
x
%
1
2RÃ
a
2
0 þ 1
À
Á
ð8:3:2Þ
Setting that the y-momentum after just after passing the finite E x region is p y ¼ p y *,
the following relation should be satisfied just at this time:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p Ã
x
2 þ p Ã
y
2 þ 1
q
À p
Ã
x ¼ RÃ
ð 8:3:3Þ
Assuming p x * is much larger than unity, R* has the minimum value for the case with
p y * ¼ 0 in the approximate form:
RÃ %
1
2p Ã
x
ð8:3:4Þ
Note that the relation (8.3.4) is the same as (8.2.14) and the role of the longitudinal
electric field is to jump the initial condition to realize (8.2.14) by the acceleration of
electrons in the x-direction with E x (<0).
In order to see the detail dynamics described above, the basic Eqs. (8.1.1) and
(8.1.2) are numerically integrated [1]. The laser amplitude is modeled as a Gaussian
pulse in the form:
a x, t
ð Þ ¼ a 0 cosξexp À
ξ
2
2 ωτ L
ð
Þ
2
"
#
ð8:3:5Þ
The laser parameters are
a 0 ¼ 10, τ L ¼ 40fs, λ L ¼ 1μm
ð8:3:6Þ
In the computation a constant longitudinal electric field is assumed for an interval of
5 μm:
b
E x ¼ À0:1a 0 142 < x μm
ð Þ < 147
ð8:3:7Þ
The potential energy difference over this electric field is quite large to change the
dephasing rate R*:
Δγ ¼ 16MeV
ð8:3:8Þ
This is about the change of p x ¼ 31, namely, the maximum energy (8.2.15) is
roughly
298
8 Chaos due to Relativistic Effect