d
dt
p ¼ þe
∂A
∂t
À v  ∇  A
ð
Þ
ð5:3:4Þ
(5.2.19) is written in the present case as
d
dt
p À eA
ð
Þ¼Àe v Á $ ∇A
½
Š
ð 5:3:5Þ
where
v Á $ ∇Aj i ¼ v j
∂A j
∂x i
ð5:3:6Þ
Since the vector potential is assumed perpendicular to the propagation direction,
(5.3.5) yields a conservation relation to the perpendicular component:
p ⊥ À eA ¼ 0
ð5:3:7Þ
where the integration constant is taken null due to the following reason. (5.3.5) is the
equation to an electron, and it has no momentum before the laser comes. We assume
that the laser intensity increases gradually with the period of many frequencies. The
quantity of (5.3.7) is conserved to the value before the laser come, where p ⊥ ¼ 0,
A ¼ 0 is satisfied.
5.3.1 Constant of Motion in Vacuum
The time evolution of the parallel momentum p k in the x-direction should satisfy the
relation from (5.3.5):
d
dt
p x ¼ Àe v y
∂
∂x
A y þ v z
∂
∂x
A z
ð5:3:8Þ
On the other hand, the energy Eq. (5.3.3) can be rewritten as
mc
d
dt
γ
ð Þ ¼
e
c
v y
∂
∂t
A y þ v z
∂
∂t
A z
ð5:3:9Þ
Taking the difference and sum of (5.3.8) and (5.3.9) leads to
180
5 Relativistic Laser-Electron Interactions