August 10, 2010 / Vol. 8, No. 8 / CHINESE OPTICS LETTERS 791
Rotary dynamics of solitons in radially periodic optical
lattices
Keya Zhou (
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), Zhongyi Guo (
HHH
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), and Shutian Liu (
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)
∗
Department of Physics, Harbin Institute of Technology, Harbin 150001, China
∗
E-mail: stliu@hit.edu.cn
Received January 4, 2010
We investigate the dynamics of strongly localized solitons trapped in remote troughs of radially periodic
lattices with Kerr-type self-focusing nonlinearity. The rotary motion of solitons is found to be more stable
for larger nonlinear wavenumbers, lower rotating velocity, and shorter radius of the trapping troughs.
When the lattice is shrunk or expanded upon propagation, the solitons can be trapped in the original
trough and move outward or inward, with their rotating linear velocity inversely proportional to the radius
of t he trapp ing troughs.
OCIS codes: 190.0190, 190.4420, 190.6135.
doi: 10.3788/COL20100808.0791.
Solitons in nonlinear s ystems play a crucial role in many
branches of nonlinear science, such as biology, solid state
physics, Bose-Einstein condensates (BECs), and nonlin-
ear optics
[1−6]
. In Kerr medium with a purely cubic non-
linear response, solitons formally exist, but in two- and
three-dimensional (2D and 3 D) cases, they are unstable
against the s patiotemporal collapse induced by the c om-
bined effect of nonlinearity and diffraction (anomalous
dispersion)
[6]
. One way to prevent colla pse is to change
the no nlinearity by considering media with qua dratic
(second-harmonic generating) nonlinearity or saturable
nonlinearity. Another way is to introduce transverse
modulation of potentials (optical lattices). In recent
years, many interests have been directed to the latter
case, and many properties of solitons that cannot be ob-
served in bulk uniform nonlinear media have been dis-
covered in transverse modulated media.
In the 2D domain, solitons have be e n investigated in
harmonic lattices
[7,8]
, hexagon lattices
[9]
, axis-symmetric
lattices
[10−16]
, and other kinds of lattices
[17−19]
. Re-
cently, axis-symmetric lattices have been the focus of
attention, where solitons are trapped either at the cen-
ter or in the remote troughs of the lattices. Different
forms of solitons have been examined, such as funda-
mental solitons
[10−12]
, dipole solitons
[13]
, ring solitons,
and necklace solitons
[14]
. An interesting phenomenon
takes place in 2D Bessel lattices or radially periodic lat-
tices. When fundamental solitons are pro bed into remote
troughs, they take a rotary motion
[10−12,15]
. However,
recent achievements in this field usually employ a funda-
mental soliton as the prob e beam, which is not formally
the exact solution of solitons trapped in remote troughs.
Solitons that are trapped in remote troughs are deter-
mined by the local distribution of the refr active index and
have different profiles fr om fundamental solitons. Their
corresponding rotary dynamics has never b een reported
befo re.
In this letter, we begin our research with the 2D Gro ss-
Pitaevskii equation (GPE)
[15]
with radially periodic po-
tential. The profiles of solitons trapped in remote troughs
of sta tionary lattices are obtained by Fourier iteration
method
[8]
. Using the solved soliton profiles as input sig-
nals, their propagation dynamics are numerically sim-
ulated by the split-step Fourier transformation method
(SSFTM). In stationary lattices, the rotating motion o f
a soliton is affected by the potential strength, nonlinear
wavenumber, rotating velocity, and radius of the troug hs.
In particular, we also investigate the rotary motion of a
soliton in dynamic lattices, which are shrunk or expanded
upon propagation. In our simulation, we use a ra ther sta-
ble soliton profile as input signal and find that solitons
are trapped in the origina l troughs and rotate around the
axis in a spiral motion upon propagation. The rotating
linear velocity is inversely proportional to the radius of
the troughs, which is determined by the two conserved
quantities of the GPE.
The model follows standard GPE in its no rmalized
form
[15]
i
∂u
∂t
+
∂
2
∂x
2
+
∂
2
∂y
2
u + V (x, y)u + |u|
2
u = 0, (1)
written for the wave function u in Cartesian coordinates
(x, y). The equation can describe the evolution of matter-
wave solitons created in BECs with resp ect to time, and
V (x, y) is the corr esponding optical lattice potential. The
dimensionless variables are chosen such that u is mea-
sured in units of the recoil energy E
r
= ¯h
2
k
2
/2m, where
k = π/ d, and d is a constant re presenting the scale of
the lattices. The temporal coordinate t and spatial co-
ordinates (x, y) a re measured in units of E
r
/¯h and d/π,
respectively. In our simulation, we consider pe riodic po-
tential in the axial direction as
V (r) = ε c os(2πr/T ), (2)
where ε is the modulation depth of the optical lattice po-
tential, and T = 2π is the dimensionless period along the
radial direction. The solitons trapped in the lattices can
be written as u(x, y, t) = U(x, y)e
jµt
, where µ is the non-
linear wavenumber of the solitons and U(x, y) denotes
the soliton profile. We use the Fourier iteration scheme
used in Ref. [6], choose a Gaussian-like initial condition,
and let the max imum of the trial function centered at
the c e nter of n
r
be the lattice troug h at po ints (n
r
T, 0).
1671-7694/2010/080791-04
c
2010 Chinese Optics Letters