Quasistable two-dimensional solitons with hidden and explicit vorticity in a medium
with competing nonlinearities
Hervé Leblond,
1
Boris A. Malomed,
2
and Dumitru Mihalache
3
1
Laboratoire POMA, UMR 6136, Université d’Angers, 2 Bd Lavoisier, 49000 Angers, France
2
Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering,
Tel Aviv University, Tel Aviv 69978, Israel
3
Department of Theoretical Physics, Institute of Atomic Physics, P.O. Box MG-6, Bucharest, Romania
Received 25 August 2004; published 15 March 2005
We consider basic types of two-dimensional 2D vortex solitons in a three-wave model combining quadratic
2
and self-defocusing cubic
-
3
nonlinearities. The system involves two fundamental-frequency FF waves
with orthogonal polarizations and a single second-harmonic SH one. The model makes it possible to intro-
duce a 2D soliton, with hidden vorticity HV. Its vorticities in the two FF components are S
1,2
= ± 1, whereas
the SH carries no vorticity, S
3
=0. We also consider an ordinary compound vortex, with 2S
1
=2S
2
= S
3
=2.
Without the
-
3
terms, the HV soliton and the ordinary vortex are moderately unstable. Within the propagation
distance z 15 diffraction lengths, Z
diffr
, the former one turns itself into a usual zero-vorticity ZV soliton,
while the latter splits into three ZV solitons the splinters form a necklace pattern, with its own intrinsic
dynamics. To gain analytical insight into the azimuthal instability of the HV solitons, we also consider its
one-dimensional counterpart, viz., the modulational instability MI of a one-dimensional CW continuous-
wave state with “hidden momentum,” i.e., opposite wave numbers in its two components, concluding that such
wave numbers may partly suppress the MI. As concerns analytical results, we also find exact solutions for
spreading localized vortices in the 2D linear model; in terms of quantum mechanics, these are coherent states
with angular momentum we need these solutions to accurately define the diffraction length of the true
solitons. The addition of the
-
3
interaction strongly stabilizes both the HV solitons and the ordinary vortices,
helping them to persist over z up to 50 Z
diffr
. In terms of the possible experiment, they are completely stable
objects. After very long propagation, the HV soliton splits into two ZV solitons, while the vortex with S
3
=2S
1,2
=2 splits into a set of three or four ZV solitons.
DOI: 10.1103/PhysRevE.71.036608 PACS numbers: 42.65.Tg
I. INTRODUCTION
Two-dimensional 2D spatial solitons with embedded
vorticity S constitute a class of topologically charged local-
ized nonlinear modes in optical media, which have recently
drawn a lot of interest, starting with the work that predicted
by means of direct simulations stable vortices with S =1 in
a model based on the 2D nonlinear Schrödinger equation
with competing self-focusing cubic and self-defocusing
quintic nonlinearities 1. Further analysis had supported this
result by showing that the spectrum of eigenmodes of small
perturbations around the vortex solitons does not contain un-
stable eigenvalues, provided that the soliton’s integral power
quadratic norm exceeds a certain minimum value in other
words, the external size of the corresponding annulus-shaped
soliton must be large enough, see details in Ref. 2. It had
also been found that higher-order vortex solitons in the 2D
cubic-quintic CQ model have their stability regions for S
=2 2a short review of the topic was given in Ref. 3, and
also for S 2 at least, up to S =5, although for S 3 the
stability region is very narrow, corresponding to the vortex
solitons with an extremely large size 4,5.
A problem hampering experimental observation of the
stable vortex solitons is that optical media, which may be
approximated by the CQ nonlinear Schrödinger NLS equa-
tion these are chalcogenide glasses 6 and some organic
substances 7, feature quite strong nonlinear loss two-
photon absorption alongside the CQ nonlinearity 8, which
can easily kill any soliton 9. For this reason, and also be-
cause it was necessary to understand how general the con-
clusion about the existence of stable spatial vortex solitons
was, this issue was further investigated in another model,
which is also based on competing nonlinearities, viz., qua-
dratic
2
alias second-harmonic-generating and self-
defocusing cubic
-
3
terms without the latter one, all the
vortex solitons in
2
media are subject to strong instability
against azimuthal perturbations, which splits them into a set
of separating zero-vorticity ZV stable solitons, as was
shown both theoretically 10 and experimentally 11. For
the first time, stable vortices with S = 1 and S =2 which were
called vortex rings, because of their annular shape in the
combined
2
:
-
3
model were found in Ref. 12. The sta-
bility was demonstrated through the calculation of stability
eigenvalues and verified by direct simulations see a brief
review in Ref. 13. Recently, this analysis was extended to
higher-order solitons, with a conclusion that, as well as in the
CQ model, the vortex rings with S 2 have their narrow
stability regions, corresponding to the rings with a very large
outer radius 14.
The
2
models involve, at least, two waves—the
fundamental-frequency FF and second-harmonic SH
ones. The soliton’s vorticity S is carried by the FF wave,
whereas the SH vorticity is 2S. In the real experiment, the
FF-SH phase matching, which is a necessary condition for
PHYSICAL REVIEW E 71, 036608 2005
1539-3755/2005/713/03660812/$23.00 ©2005 The American Physical Society 036608-1