In Fig. 6.8c, the transverse electron momentum p z is plotted same as the longitudinal one. It is seen that the bulk electrons are oscillating in the z-direction even
after the passage of laser pulse. This also indicates that the low-frequency electromagnetic waves are left behind the laser pulse with large amplitudes to keep the
oscillation of p z . In addition, some electrons accelerated by the wake field are
rotating along the propagation axis by the fields of condensed low-frequency
electromagnetic waves.
6.3 Ponderomotive Force in Relativistic Field
The ponderomotive force (PM force) is the force appearing after the time average of
the force by oscillating field. This force is due to the inhomogeneity of the field
strength and is very important to plasma dynamics interacting with short pulse and
spatially lasers. The PM force is obtained as follows in the case of electromagnetic
fields.
Begin with the equation of motion of an electron shown in (5.3.4):
dp
dt
¼
∂p
∂t
þ v Á ∇p ¼ e
∂Α
∂t
À
e
mγ
p  ∇  Α
ð
Þ
ð6:3:1Þ
Since the generalized momentum P defined by
P ¼ p À eA
is conserved in the total system of an electron and fields, A in the second term in
(6.3.1) can be replaced with p. Then, (6.3.1) reduces to
∂p
∂t
¼ e
∂A
∂t
À
1
mγ
p Á ∇p À p  ∇  p
ð
Þ
½
Š
ð 6:3:2Þ
Using a mathematical relation:
p Á ∇
ð
Þp ¼
1
2
∇ p
j j
2 þ p  ∇  p
ð
Þ,
ð6:3:3Þ
(6.3.2) reduces:
∂p
∂t
¼ e
∂A
∂t
À
1
2m
γ
À1
∇ p
j j
2 :
ð6:3:3aÞ
Taking the time average of (6.3.3a) over laser oscillation < > to remain only slowly
varying terms, the following simple relation is obtained:
6.3 Ponderomotive Force in Relativistic Field
217