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Scientific Report 2007-2009
Theoretical physics
T20. Optical solitons in resonant interactions of three waves
In glass nonlinearity is cubic (Kerr effect) and solitons
result from balance between dispersion (or diffraction)
and nonlinear self-focusing. More recently both theoretical
and experimental interest has been attracted by
soliton propagation in media with quadratic nonlinearity
(as in KTP crystals). In these media the most important
and applicable effects arise in the resonant interaction
of three waves, 3WRI. This interaction is modelled by
a system of three dispersionless quadratic nonlinear partial
differential equations for the envelopes of three quasi
monochromatic plane-waves. Energy exchange between
these three waves is possible because of the resonce condition
ω 1 + ω 2 = ω 3 . These equations are integrable and
their analytic investigation is made possible by the powerful
tools of spectral theory. Since dispersion is missing,
the mechanism of soliton formation is quite different
from that in cubic material. In this case is rather
the mismatch of group velocities and nonlinearity which
gives rise to soliton propagation.
It is well known that parametric three-wave mixing
provides a means of achieving widely tunable frequency
conversion of laser light. Moreover the frequency conversion
of short (bright) pulses may be significantly enhanced
by means of optical solitons. Indeed, the collision
of two bright input (soliton) pulses at different frequencies,
with proper duration and input power, leads to a
time-compressed pulse at the sum-frequency. However
such pulse is unstable, since it rapidly decays into two
time-shifted replicas of the same input pulses, with obvious
limitation of the applicability of this technique to
frequency conversion. In this context, a substantial advancement
started in our group in Rome with the discovery
of a new multi-parametric class of soliton solutions
of the 3WRI model. The novelty of these solitons
is that they describe a triplet made up of two short (localised)
pulses and a cw background. Because of the persistent
interaction with the background, the two b! right
pulses propagate with dispersion whose balance with the
quadratic nonlinearity causes a quite rich, and non standard,
phenomenology of soliton behaviour. The distinction
of these solitons with respect to those previously
known is emphasised by referring to them as boomerons
and trappons.
The most elementary solitons of this new family are
bright-bright-dark triplets which travel with a common,
locked velocity. Their velocity is different from any
of the three characteristic group velocities. A soliton
(simulton) of this type is stable if its velocity V is
higher than a critical value V cr . Unstable simultons
move with velocity which is lower than V cr but then
eventually decay in an higher velocity stable soliton and
a bump of the background moving with its own characteristic
velocity. This implies that the initial and final
velocities of unstable simultons are different from each
other, namely they are accelerated. The time reverted
process describes the excitation of a stable simulton by
absorption of a background bump. These are analytic
solutions of the 3WRI equations, the dynamical process
being the boomeron solution. However the physical
process consists of an excitation followed by decay (see
Fig.1). Simultons may also couple together according
Figure 1: Numerical double boomeron process.
to their phase relations. For instance two in-phase
simultons attract each other in a bound state in an
even richer coherent structure (see Fig.2). A variety of
Figure 2: Two in-phase simultons with the same velocity.
potential applications are at hand by using the soliton
behaviours of Bright-Bright-Dark triplet. For instance,
an ultra short pulse (signal) interacting with the cw
background (pump) generates a sum-frequency short
bright pulse whose intensity, width and velocity can be
controlled in a stable and efficient way by varying the
background intensity. Moreover the trappon solution, a
triplet of two bright and a dark pulses which are locked
together to periodically oscillate, can lead to device a
way to generate high-repetition rate pulse trains whose
applicability and interest is in a broad range of domains
[1]. On the experimental side, the first observation of
solitonic decay in the case of three bright pulses has
been reported quite recently [2] as the first step towards
the observation of purely boomeronic and trapponic
processes.
References
1. F. Baronio et al., IEEE J. Quant. Elect 44, 542 (2008).
2. F. Baronio et al., Opt. Expr. 17, 13889 (2009).
Authors
F. Calogero 1 , A. Degasperis
Sapienza Università di Roma 43 Dipartimento di Fisica