b x t
ð Þ ¼
a
2
0
4
ϕ þ
1
2
sin 2ϕ
b y t
ð Þ ¼ a 0 sin ϕ
ð Þ
ð5:3:23Þ
Inserting the phase, we know that the electron is moving in the x-direction:
b x t
ð Þ ¼
a
2
0
4 þ a 2
0
b t þ
1
2
sin 2ϕ
ð5:3:23aÞ
which provides a constant drift velocity V d in the x-direction with the form
b
V d ¼
a 0
2
a 0
2 þ 4
ð5:3:24Þ
The electron velocities are easily obtained in the forms. (5.3.22b) can be rewritten
with use of (5.3.18) to the form:
b v x ¼
a
2
2γ
¼
1
2
a
2
1
2
a
2
þ 1
b v y ¼
a
γ
¼
a
1
2
a
2
þ 1
ð5:3:25Þ
It is clear that the time average velocity in x-direction is equal to (5.3.24).
In Fig. 5.3, the time-averaged drift velocity V d in the x-direction (red) and the
amplitude of the quivering velocity in y-direction (blue) are plotted as functions of
laser strength a 0 . Typical orbits with a 0 much smaller or larger than the unity are
1.0
0.8
V/C
a 0
0.6
0.4
0.2
0
2
4
6
8
1 0
Oscillation velocity in y-direction (E force)
Drift velocity in x-direction (vxB force)
Fig. 5.3 Time-averaged drift velocity V d of (5.3.24) (blue) and the oscillation amplitude in
y-direction of (5.3.25) (red) as a function of the field strength a 0
184
5 Relativistic Laser-Electron Interactions