Y
2
À ω
2
0 þ 2c
2 k
2
y
Y À D ¼ 0,
ð6:5:25Þ
where
D ¼
a
2
0
8
ω
2
p0 À c
2 k
2
y
a
2
0
8
ω
2
p0 þ c
2 k
2
y
ð6:5:26Þ
It is clear that (6.5.25) has a negative solution Y ¼ δ
2 < 0 for D > 0. The negative
solution means the frequency shift δ has pure imaginary and the system (6.5.22)
becomes unstable.
The condition for the two waves unstable is the condition of the filamentation
instability. Two waves consist of the standing wave in the y-direction as
a 1 þ a 2 ) 2 exp δ
j jt
ð Þcos k y y
À Á
cos k 0 x À ω 0 t
ð
Þ
ð 6:5:27Þ
The growth of the amplitude is due to the energy flow of the fundamental wave,
namely, the plane homogeneous wave of the laser which is going to be modified
sinusoidally in the perpendicular direction with exponential increase of its
amplitude.
The condition of D > 0 provides a new condition for the beam size to be selffocusing:
k y <
a 0
2
ffiffi ffi
2
p
1
ρ s
ð6:5:28Þ
With increase of the laser intensity and/or density of plasmas, the critical beam size
(6.5.28) gets smaller. It is informative to compare this critical beam size for
filamentation instability with that obtained from (6.5.11). The condition that the
relativistic mass correction is stronger than the diffraction effect gives the critical
radius in the form from (6.5.11):
1
ffiffi ffi
2
p
R
<
a 0
2
ffiffi ffi
2
p
1
ρ s
ð6:5:29Þ
It is a good coincidence that by replacing an effective wavenumber k y with 1=
ffiffi ffi
2
p
R,
rough evaluation in (6.5.11) coincides with the precise analytic result.
For the case the wavenumber k y is much smaller than the critical value (6.5.28),
we can obtain the growth rate of such filamentation instability, |δ| γ FI in an
approximate form:
228
6 Relativistic Laser Plasma Interactions
2
À ω
2
0 þ 2c
2 k
2
y
Y À D ¼ 0,
ð6:5:25Þ
where
D ¼
a
2
0
8
ω
2
p0 À c
2 k
2
y
a
2
0
8
ω
2
p0 þ c
2 k
2
y
ð6:5:26Þ
It is clear that (6.5.25) has a negative solution Y ¼ δ
2 < 0 for D > 0. The negative
solution means the frequency shift δ has pure imaginary and the system (6.5.22)
becomes unstable.
The condition for the two waves unstable is the condition of the filamentation
instability. Two waves consist of the standing wave in the y-direction as
a 1 þ a 2 ) 2 exp δ
j jt
ð Þcos k y y
À Á
cos k 0 x À ω 0 t
ð
Þ
ð 6:5:27Þ
The growth of the amplitude is due to the energy flow of the fundamental wave,
namely, the plane homogeneous wave of the laser which is going to be modified
sinusoidally in the perpendicular direction with exponential increase of its
amplitude.
The condition of D > 0 provides a new condition for the beam size to be selffocusing:
k y <
a 0
2
ffiffi ffi
2
p
1
ρ s
ð6:5:28Þ
With increase of the laser intensity and/or density of plasmas, the critical beam size
(6.5.28) gets smaller. It is informative to compare this critical beam size for
filamentation instability with that obtained from (6.5.11). The condition that the
relativistic mass correction is stronger than the diffraction effect gives the critical
radius in the form from (6.5.11):
1
ffiffi ffi
2
p
R
<
a 0
2
ffiffi ffi
2
p
1
ρ s
ð6:5:29Þ
It is a good coincidence that by replacing an effective wavenumber k y with 1=
ffiffi ffi
2
p
R,
rough evaluation in (6.5.11) coincides with the precise analytic result.
For the case the wavenumber k y is much smaller than the critical value (6.5.28),
we can obtain the growth rate of such filamentation instability, |δ| γ FI in an
approximate form:
228
6 Relativistic Laser Plasma Interactions