Beam Trapping and Filaments 209
nonlinear polarization. The condition for the existence of a temporal soliton
is that (anomalous) dispersion
ik
2
∂
2
∂t
2
balances self-temporal lensing (positive
SPM)
in
2
k
n
0
|E|
2
˜
E. Normal dispersion and a negative SPM can also lead to soliton
solutions [71].
3.9. BEAM TRAPPING AND FILAMENTS
Once the beam power is sufficient for self-focusing to overcome diffraction,
the beam collapses to a point. In general, after the beam has collapsed, it diffracts.
However, numerous experiments have shown self-guiding of high peak power
infrared femtosecond pulses through the atmosphere [72–77]. Similar observa-
tions were made in the UV [78, 79]. After reaching the focus, the light appeared
to trap itself in self-induced waveguides or “filaments” of the order of 100 µm
diameter. Before addressing problems specific to ultrashort pulses we will discuss
a steady state model to illustrate the possibility of beam self-trapping.
3.9.1. Beam Trapping
We start with the time-free wave equation
∂
∂z
+
i
2k
∂
2
∂x
2
+
∂
2
∂y
2
˜
E(x, y, z) =−ik
NL
˜
E(x, y, z). (3.178)
For the nonlinear propagation constant on the right-hand side we assume a non-
linear refractive index because of a Kerr nonlinearity and a contribution of next
order.
k
NL
=
ω
c
n
2
|
˜
E|
2
+ n
3
|
˜
E|
3
(3.179)
A physical system that can give rise to a negative n
3
will be introduced later.
Ageneral solution to Eq. (3.178) is only possible by numerical means. To illustrate
the possibility of beam trapping we will analyze the term on the right-hand
side of the wave equation near in the vicinity of the beam center. Assuming a
Gaussian beam profile,
˜
E = E
0
(w
0
/w) exp(−r
2
/w
2
)), over a propagation distance
z, the medium introduces a phase factor
φ
NL
(r) = k
NL
z =
ω
c
n
2
E
2
0
w
2
0
w
2
e
−2r
2
/w
2
+ n
3
E
3
0
w
3
0
w
3
e
−3r
2
/w
2
z. (3.180)
210 Light–Matter Interaction
The curvature of the r dependent phase on-axis determines the focusing char-
acteristics of a slice of thickness z. For n
3
= 0 this is the phase factor that
was discussed in Section 3.8.1 and was found responsible for self-focusing.
The curvature of the phase term in the vicinity of the beam center is
d
2
dr
2
φ
NL
(r)
r≈0
=−Q
1 + E
0
w
0
n
3
wn
2
(3.181)
where Q = 4ω
n
2
˜
E
2
0
w
2
0
z/(cw
4
). For n
2
> 0 as is the case in most materials and
n
3
< 0 we have the situation that the term in brackets can change sign depending
on the ratio n
3
/(n
2
w) for given input beam (
˜
E
0
w
0
). For n
2
w > (<) n
3
the material
will act like a positive (negative) lens. A positive lens tends to decrease w on
propagation until at some point the sign of the phase term reverses leading to
negative lensing. This in turn increases w until the process is reversed again.
This suggests the possibility of a periodically changing beam diameter (trapped
beam) even if diffraction is included. The effect of the latter is that the phase
curvature should have a certain positive value depending on w before the beam
actually contracts. Similar beam trapping can be expected from contributions that
are of order m > 2 if the sign of n
m
is negative. The nonlinear refractive indices
have their origin in nonlinear susceptibilities of order m +1. The Kerr effect, χ
(3)
producing a nonlinear index n
2
, is one example, which we discussed in detail
previously.
Let us now briefly describe a physical system that can produce a negative n
3
.
Let us assume that the beam propagates through a gas that can be ionized by a
three photon absorption. The free electrons (density N
e
) can recombine with the
positive ions and a steady state will be reached.
d
dt
N
e
= σ
(3)
|
˜
E|
6
N
0
− β
ep
N
2
e
= 0 (3.182)
Here σ
(3)
is a three photon absorption cross section, N
0
is the number density
of atoms, and β
ep
is the two-body recombination constant. From Eq. (3.182) we
can obtain the steady state density of free electrons as a function of the laser field
N
eq
=
-
σ
(3)
N
0
β
ep
|
˜
E|
3
. (3.183)
The (small) change of the refractive index ˜n associated with the laser generated
free electrons can be estimated with the Drude model:
˜n
2
= (1 +˜n)
2
≈ 1 +2˜n = 1 +
ω
2
p
ω
2
1 − i
γ
ω
. (3.184)
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