Solitons in Nonlocal Media

1.3 Nonlocality
(6), according to the propagation coordinate taken into account 1 . Moreover, a soliton
can be a bright spot in a dark background or a light dip in a uniform background;
beams of the former kind are named bright solitons, those of second are dark solitons
(21).
Optical temporal solitons are pulses which maintain their shape in time in nonlinear
guides owing to the balance between broadening, due to the unavoidable dispersion,
and nonlinear self-phase modulation. They have been extensively investigated owing
to their possible applications in fiber optics, in order to improve the bit-rate (22).
Conversely, spatial solitons are nonlinear waves stationary with respect to time; they
do not change their spatial profile as they propagate (5; 23; 24). As it is well known,
in linear homogeneous media electromagnetic waves diffract, i.e. their transverse width
increases along propagation. In some nonlinear media, as I discussed above referring
to the Kerr effect, beams are capable (for large enough input powers) to self-focus, i.e.
to form a lens. When this two counter-acting effects are perfectly balanced, a soliton
is formed. Because of this formation mechanism, soliton shape and power are strongly
dependent on the specific nonlinearity. For example, in Kerr media only solitons in
(1+1)D 2 , i.e. in slab nonlinear waveguides, are stable; in (2+1)D they are unstable,
i.e. solitons are destroyed by beam collapse. To obtain stable solitary propagation in
(2+1)D it is necessary to exploit some other kind of nonlinearities, for example sat-
urable or nonlocal ones (6).
Finally, solitons with profiles containing more than a local maximum, known as higher
order solitons in analogy with higher-order modes of linear guides, have been demon-
strated as well (5).
In this thesis I will focus on bright spatial solitons.
1.3 Nonlocality
Generally speaking, a medium is nonlocal when its response to an excitation is not
null even in points where the excitation is zero, i.e. the Green function has a finite
1 Actually, there is a third kind of soliton called bullet, nonlinearly self-localized in both space and
time.
2In this notation the first and second number are the transverse and propagation coordinates,
respectively.
3