Solitons in Nonlocal Media

1.3 Nonlocality
specified, from here on I will consider homogeneous nonlinear media, i.e. with ∆n not
explicitly dependent on space 1 . Eq. (1.2) becomes
∂A
2ik0n0
∂s + ∇2⊥ A + k2 02n0∆n(I)A = 0 (1.3)
This is the nonlinear Schröedinger equation (NLSE) for a generalized nonlinearity,
widely used in modeling spatial solitons. Assuming a linear relationship between the
intensity I and the index perturbation ∆n (for example, thermo-optic media and liquid
crystals in limited range of powers), and supposing that ∆n on a certain plane normal
to s depends on intensity in that plane, 2 I get
∆n =
I(r ′ ⊥ )G(|r⊥ − r ′ ⊥ |)dS′
(1.4)
where dS ′ and r⊥ are the infinitesimal area element and position vector on the
transverse plane, respectively, and G(|r⊥ − r ′ ⊥ |) is the Green function for the material
(32). I introduced a Green function depending only on the distance between excitation
and effect, that is, an infinitely extended medium. Finite geometries will be discussed
in chapter 3.
A nonlinear material which is described by eqs. (1.3) and (1.4) is nonlocal Kerr. In
local Kerr media I have ∆n = n2I and eq. (1.3) turns into the classical NLSE, which
is integrable and supports the fundamental soliton with a sech profile (6).
Different ranges of nonlocality have been discussed in literature: from high (31; 39; 50;
51; 52; 53; 54) to weak (32; 55).
1.3.1 Strong Nonlocality
Let me begin with the highly nonlocal case and, for the sake of simplicity, explore its
features in a one dimensional geometry. Expanding G(x − x ′ ) in eq. (1.4) in a Taylor
series around the point x = x ′3 , I get:
1 I note that the nonlinear index perturbation is varying in space due to its dependence on I.
2 According to the hypothesis, ∆n is governed by L(∆n) = I with L a certain linear differential
operator; in case of solitary propagation I have that all the s derivatives become null due to the
invariance in propagation, hence the index perturbation on a plane normal to s depends only on
intensity computed on that plane.
3 I implicitly assume that the Green function is derivable in x = x ′ ; this is not always true, as it
will be shown in chapter 3.
5