γ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b p
2
x þ b p
2
⊥ þ 1
q
ð5:3:14Þ
Inserting (5.3.13) into (5.3.14), we obtain the following relation:
γ t
ð Þ ¼
1
2α
a
2
þ α
2
þ 1
Â
Ã
ð5:3:14aÞ
Eliminating γ from (5.3.13) and (5.3.14), the following simple relation is obtained:
b p x ¼
1
2α
1 À α
2
þ b p
2
⊥
h
i
ð5:3:15Þ
5.3.3 Electron Motion in Vacuum
Including any polarization of laser fields, the following expression to A is usually
used:
A ϕ
ð Þ ¼ 0, δA 0 cosϕ, 1 À δ
2
À
Á 1=2 A 0 sinϕ
h
i
ð5:3:16Þ
where ϕ is the phase of the laser:
ϕ t, x
ð Þ ¼ ωt À kx
ð5:3:17Þ
The linear polarization to y or z direction is given for δ ¼ 1, À1 or 0. Right or left
rotating circularly polarized lasers is given for δ ¼ 1=
ffiffi ffi
2
p
, À 1=
ffiffi ffi
2
p
.
It is useful to know the following relation in the frame of the moving electron x(t):
d
d b t
ϕ t, x t
ð Þ
½
Š¼1 À
db x
d b t
¼ 1 À
b p x
γ
¼
α
γ
ð5:3:18Þ
In deriving (5.3.18), (5.3.13) is used. Taking time average of (5.3.18), it is shown
that the time-averaged frequency in the moving electron frame is given by
ω
0
¼
α
γ
h i
ω
ð5:3:19Þ
We consider the two cases with different α value. In the case of a free electron, α ¼ 1,
and the other case is for α ¼ hγi. As clear from (5.3.19), an electron feels down-shift
of frequency, and it keeps moving to the laser propagation direction. In the latter
case, the average position of an electron has no move as clear from (5.3.19), and this
case is appropriate for many electrons in plasmas as will be explained later.
182
5 Relativistic Laser-Electron Interactions