Self-Focusing 205
3.8. SELF-FOCUSING
The nonlinearities of a medium affect both the temporal and spatial depen-
dence of the electric field of the light. In the previous sections we have avoided
this difficulty by assuming a uniform beam profile or neglecting nonlinear
space–time coupling effects. However, any nonlinear interaction strong enough
to affect the pulse temporal profile will also affect its transverse profile. One
example is SHG, slightly off-exact phase matching condition (large nonlinear
phase shifts), or at large conversion efficiencies. As sketched in the inset of
Fig. 3.9, an initially Gaussian temporal profile will be depleted predominantly in
the center, resulting in a flattened shape. The same interaction will also transform
an initially Gaussian beam into a profile with a flat top.
In this section we will consider as an important example the problem of
self-focusing. The intensity-dependent index of refraction causes an initially
collimated beam to become focused in a medium with n
2
> 0. It is the same
intensity dependence of the refractive index that causes SPM of a fs pulse, as we
have seen in the previous section.
3.8.1. Critical Power
The action of a nonuniform intensity distribution across the beam profile on a
nonlinear refractive index results in a transverse variation of the index of refrac-
tion, leading either to focusing or defocusing. Let us assume a cw beam with a
Gaussian profile I = I
0
exp(−2r
2
/w
2
), and a positive ¯n
2
, as is typical for a non-
resonant electronic nonlinearity. The refractive index decreases monotonically
from the beam center with increasing radial coordinate. One can define a “self-
trapping” power P
cr,1
as the power for which the wavefront curvature (on axis)
because of diffraction is exactly compensated by the change in the wavefront
curvature because of the self-lensing over a small propagation distance z.We
assume that the waist of the Gaussian beam is at the input boundary of the non-
linear medium (z = 0). Within the paraxial approximation, diffraction results in a
spherical curvature of the wavefront at a small distance z from the beam waist:
ϕ
diff
(z) =−
k
z
2ρ
2
0
r
2
, (3.167)
where ρ
0
is the Rayleigh range. This result is obtained by approximating
1/R ≈ z/ρ
2
0
in Eq. (1.181). The action of the nonlinear refractive index results
206 Light–Matter Interaction
in a radial dependence of the phase after a propagation distance z:
ϕ
sf
(r, z) =−¯n
2
2π
λ
zI
0
e
−(2r
2
/w
2
0
)
≈−¯n
2
2π
λ
zI
0
1 − 2
r
2
w
2
0
, (3.168)
where the last equation is an approximation of the wavefront near the beam
center. This equation follows from Eq. (3.149) after replacing n
2
E
2
0
by ¯n
2
I(r).
The input beam has the critical power P
cr,1
=
2πrI(r)dr when the radial parts
of Eqs. (3.167) and (3.168) compensate each other:
P
cr,1
= I
0
πw
2
0
2
=
λ
2
8πn
0
¯n
2
(3.169)
where we have made use of ρ
0
= πw
2
0
n
0
/λ. One says that the beam is self-
trapped because neither diffraction nor focusing seems to occur. This value of
critical power is also derived by Marburger [66] by noting that the propagation
equation is equivalent to that describing a particle moving in a one-dimensional
potential. The condition for which the potential is “attractive” (leads to focusing
solutions) is P ≥ P
cr,1
. One can define another critical power P
cr,2
as the power
for which the phase factor on-axis of the Gaussian beam, arctan(z/ρ
0
), exactly
compensates the nonlinear phase shift ¯n
2
I
0
:
P
cr,2
=
λ
2
4πn
0
¯n
2
. (3.170)
This second value of the critical power is also obtained by assuming that the
beam profile remains Gaussian and deriving an expression for “scale factor”
f (z) = w(z)/w
0
as a function of distance z [67, 68]. The function f (z) reaches
zero after a finite distance if the power exceeds the value P
cr,2
.
Another common approach to defining a self trapping power is to approximate
the radial beam profile by a flat top of diameter d [8]. The refractive index inside
the tube of diameter d is n = n
0
+¯n
2
I
0
. The critical angle for total internal
reflection, α, is determined by sin α = n
0
/(n
0
+¯n
2
I
0
). The beam is trapped inside
the tube if the diffraction angle θ
d
≈ 1. 22λ/(2n
0
d) is equal to θ
cr
= π/2 − α.
From this condition and using the fact that ¯n
2
I
0
n
0
one can derive the critical
power P
cr,3
= I
0
πd
2
/4:
P
cr,3
=
(1. 22)
2
πλ
2
32n
0
¯n
2
(3.171)
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