212 Light–Matter Interaction
end we insert the ansatz of a Gaussian beam profile
˜
E = E
0
[w
0
/w(z)]exp(−r
2
/w(z)
2
)
and assume that the Gaussian beam shape is maintained throughout the propa-
gation (focusing). The beam waist w
0
is at the sample input (z = 0). Within the
frame of paraxial optics we approximate the nonlinear term by
n
2
|
˜
E|
2
≈ n
2
E
2
0
w
2
0
w
2
(z)
1 − 2
r
2
w
2
(z)
. (3.186)
Sorting the powers of r and setting the prefactors to zero results in a second-order
differential equation for the beam waist:
d
2
dz
2
w(z) =−
4
k
2
P
P
cr
− 1
1
w
3
(z)
, (3.187)
where P is the (constant) beam power and P
cr
is the critical power defined as
P
cr,1
in Eq. (3.169). The solution to this differential equation is
w
2
(z) = w
2
0
−
w
2
0
ρ
2
0
P
P
cr
− 1
z
2
(3.188)
where ρ
0
= kw
2
0
/2 is the Rayleigh range. Provided that P > P
cr
, the beam
collapses at a distance equal to the self-focusing length z = z
SF
, where
z
SF
=
ρ
0
√
P/P
cr
− 1
. (3.189)
3.9.2. Ultrashort Pulse Self-Focusing
An exact treatment of short pulse self-focusing is complex as it involves the
inclusion of many different nonlinear effects combined with propagation effects.
The general approach is to start with Eq. (3.97) and specify the nonlinear polar-
ization of the material in which the pulse propagates. For example, by specifying
the nonlinear polarization as the Kerr effect and the Raman effect, one arrives
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