Solitons in Nonlocal Media

3.3.3 Highly Nonlocal Case
3.4 Soliton Oscillations in a Finite-Size Geometry
Let me change prospective and consider an infinitely narrow beam, i.e. η(y) = δ(x −
〈x〉). From eq. (3.62) I obtain
F m X = W0(〈x〉) (3.66)
Physically, beams much narrower than the width of the nonlinear response, i.e. in
the highly nonlocal regime, are affected only by the derivative of the nonlinear index
profile computed on the beam center. I stress how eq. (3.66) keeps its validity whenever
Wn terms for n ≥ 2 are negligible with respect to W0.
I can apply eq. (3.65) under hypotheses employed for its derivation (see 3.3.2); potential
is then given by
When c 1 0
V m X (〈x〉,s) = − 1
2 c1 0 〈x〉 2
(3.67)
is constant along s, the potential exerted on the beam does not change
with 〈y〉 n , i.e. the beam width does not affect its trajectory. Thus, the beam moves
along a sinusoidal trajectory around x = 0, with an amplitude determined by initial
conditions and a period Λ = 2π
k
c 1 0
.
3.4 Soliton Oscillations in a Finite-Size Geometry
In this section I apply the theory developed in the latter section to investigate beam
motion in nonlocal media of finite size. I will focus on the four cases presented in section
3.2, considering Gaussian shapes and their evolution in the presence of an equivalent
force given by eq. (3.66) due to the different distance from boundaries. Such results
will be confirmed by numerical simulations based on the NNLSE, and by experiments
in NLC.
3.4.1 Poisson 1D
Applying eqs. (3.56), (3.66) and (3.61) to the nonlinear index perturbation (3.8) in the
normalized transverse unit ξ = x/a and considering |A| 2 = Ce −ξ2 /ω 2
62
with C = k2 0 2PLZ0
√ πωaβn0