d
dt
ξ ¼ 1 À v x ¼
γ À p x
ð
Þ
γ
¼
α
γ
ð9:2:1Þ
This is rewritten with the proper time:
d
dτ
ξ ¼ α
ð9:2:2Þ
Note that the canonical momentum in the y-direction is still conserved.
For the case where the electron moves to the next filament at the time “n,” the
y-momentum should satisfy the relation at time t from (8.4.8) just after the jump at n:
p y ¼ p
n
y þ a 0 sinξ À sinξ
n
ð
Þ
ð 9:2:3Þ
Inserting a(t,x) to (8.4.9), the following equation is obtained:
d
dτ
p x ¼ Àa 0 p y cosξ
ð9:2:4Þ
Using (9.2.2) and (9.2.3), (9.2.4) is modified like
d
dξ
p x ¼ À
a 0
α
p y cosξ ¼ À
a 0
α
p
n
y þ a 0 sinξ À sinξ
n
ð
Þ
h
i
cosξ
ð9:2:5Þ
Integrating (9.2.5) for a very short time interval across the time of the event “n,”
(9.2.5) yields
120
Contours of transverse electric field
I initial = 10
19 W/cm
2
y(c/ω
0 )
x(c/ω 0 )
0
0
9 0
Fig. 9.1 A plot of
transverse laser electric field
in real space, where the laser
must propagate through
under-dense plasma before
reaching critical density.
Notice the combination of
both self-focusing and
filamentation. [Figure 12 in
Ref. 2]
9.2 Stochastic Heating by Laser Filamentation
335