208 Light–Matter Interaction
3.8.2. The Nonlinear Schrödinger Equation
Let us consider the propagation of a laser beam along the direction z,ina
medium characterized by a linear index of refraction n, a third-order nonlinear
polarization and a linear loss–gain coefficient α [cf. Eq. (3.141)]. The nonlinear
polarization
˜
P
(3)
NL
=
3
8
0
χ
(3)
|
˜
E|
2
˜
Ee
i(ω
t−kz)
. (3.174)
is to be substituted into the wave equation (1.67), with the field given by Eq (1.83).
If we assume a steady state condition (no time dependence) the equation
describing the spatial dependence of the electric field is:
∂
∂z
+
i
2k
∂
2
∂x
2
+
∂
2
∂y
2
+ i
3ω
2
8c
2
k
χ
(3)
|E|
2
−
α
2
˜
E = 0 (3.175)
The prefactor in front of the nonlinear term can also be written as
3ω
2
χ
(3)
/(8c
2
k
) = n
2
k
/n
0
. One recognizes in Eq. (3.175) a three-dimensional
generalization of the nonlinear Schrödinger equation [69]:
i
∂ψ
∂z
− a
∂
2
ψ
∂x
2
− b|ψ|
2
ψ + cψ = 0. (3.176)
The last term of Eq. (3.175) is a linear gain or absorption associated with the imag-
inary part of the linear index of refraction. In one dimension, these equations were
shown by Zakharov and Shabat [70] to have steady state solutions labelled “soli-
tons.” These solitons correspond to a balance between the self-focusing and the
diffraction. The physical reality is however more complex than can be included
in the nonlinear Schrödinger Eq. (3.176). Indeed, once the nonlinearity exceeds
the threshold to overcome diffraction, the beam collapses to a point (see next
section). To obtain a dynamically stable solution in the transverse dimension, it
is necessary to include a higher-order nonlinearity in the polarization to prevent
this collapse, as will be demonstrated.
If we consider a short pulse propagating as a plane wave through an infinite
medium with the nonlinear polarization of Eq. (3.174), the temporal evolution
of the field is given by a similar nonlinear Schrödinger equation:
∂
∂z
−
ik
2
∂
2
∂t
2
+ i
n
2
k
n
0
|E|
2
−
α
2
˜
E = 0, (3.177)
where the independent variables are now z and t. The soliton solution to
Eq. (3.177) results from a balance between dispersion and SPM caused by the
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