COL 11(9), 090801(2013) CHINESE OPTICS LETTERS September 10, 2013
Self-trapping and oscillation of quadruple beams in high
band gap of 2D photonic lattices
Shiqiang Xia (
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), Daohong Song (
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), Liqin Tang (
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∗
,
Cibo Lou (
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), and Yigang Li (
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The MOE Key Laboratory of Weak Light Nonlinear Photonics, TEDA Applied Physics School and School of Physics,
Nankai University, Tianjin 300457, China
∗
Corresponding author: tanya@nankai.edu.cn
Received March 14, 2013; accepted June 28, 2013; posted online September 3, 2013
Higher-band self-trapping and oscillation (rotation) of nonlinear quadruple beams in two-dimensional (2D)
square photonic lattices are numerically demonstrated. Under appropriate conditions of nonlinearity, a
quadruple-like beam can self-trap into localized modes that reside in the second Bragg reflection gap
through single-site excitation. By changing the initial orientation of th e incident quadruple beam related
to the lattices, periodic oscillations of t he localized quadruple mo de may be obtained. The localized
quadruple state becomes a rotating doubly charged optical vortex (DCV) during rotation and should
undergo charge-flipping when t he rotating direction is reversed.
OCIS codes: 080.1238, 190.4420, 050.4865.
doi: 10.3788/COL201311.090801.
Linear and non-linear light propag ation in perio dic pho-
tonic lattice has attrac ted strong research interest in re-
cent years, and many fascinating light behaviors
[1]
have
been demonstrated to contain a physical essence beyond
optics, such as Anderson localization
[2]
, Bloch oscillation,
and Ze ner tunneling
[3]
. Photonic lattices se rve as an ideal
platform for exploring various fundamental issues in dis-
crete sy stems, and exciting phenomena in new lattice
settings, including super lattices
[4]
, ionic-type photonic
lattices
[5]
, and three-dimensional photonic lattices
[6,7]
,
have been demonstra ted. In a linear photonic lattice,
light propagation is modulated in acco rdance with the
Floquet-Bloch theorem
[8]
. The continuous spatial trans-
mission spectrum is split into s e parate bands by forbid-
den gaps at the b oundaries of the Brillouin zone due to
Bragg reflections of the periodic potential. In the pre s-
ence of appropriate nonlinearity, discrete diffractions can
be balanced, with light self-trapping into localized sta tes
as la ttice solitons. Many types of localized modes re-
siding in different band gaps have been reported e xper-
imentally or theoretically in optically-induced photonic
lattices on the basis of various band-gap s tructures, such
as fundamental solitons
[9]
, dipole solitons
[10−13]
, vortex
solitons
[14,15]
, bandgap surface vortex solitons
[16]
, higher-
band vortex solitons
[17]
, and embedded-soliton trains
[18]
,
to name only a few. Rotating properties of la ttice solitons
have also been demonstrated
[19−23]
. For instance, in op-
tically induced periodic ring lattices, the soliton rotation
in different lattice rings ca n be controlled by imposing an
initial transverse momentum on the soliton
[21]
. Doubly
charged vortices (DCVs)
[22]
and dipole solitons
[23]
have
been reported to experience rota tions in a square pho-
tonic la ttice, which is especially interesting because the
dipole soliton itself initially car ries no angular momen-
tum. Nevertheless, all of the solitons mentioned above
are discrete solitons or gap solitons existing in the total
internal reflection g ap and the fir st Bragg reflection band
gap. Therefore, whether or not a localized state residing
in higher ba nd gaps can be excited, as well as the dy-
namics of such excitation, must be investigated.
In this letter, we firstly calculate the disp ersion relation
(band structure) of two-dimensional (2D) square lattices
and then determine the possibilities for localized quadru-
ple states that reside in the second Bragg reflectio n band
gap by inputting a quadruple-like beam throug h sing le-
site excitation and investigating its rotational proper-
ties by analy zing its linear and nonlinear propagation in
different orientations relative to the lattices.
Beam propagation through an optically induced pho-
tonic lattice in a biased photorefractive crystal is gov-
erned by the paraxial non-linear Schr¨odinger equation
[24]
:
i
∂ψ
∂z
+ D
∂
2
ψ
∂x
2
+
∂
2
ψ
∂y
2
−
γψ
1 + I
b
(x, y) + ζ |ψ|
2
= 0,
(1)
where ψ is the slowly varying complex ampli-
tude of the probe beam, ζ = 0, 1 represent lin-
ear and nonlinear conditions, respectively, I
b
=
I
g
{cos [π (x + y)/d] + cos [π (x − y)/d]}
2
is the lattice
potential translated into a change in the refractive in-
dex ∆n through the non-linearity in a photorefractive
crystal, I
g
is the lattice peak intensity in units of back-
ground irradiance, and d = 23 µm is the lattice spac ing.
The lattice is written in this fashion because it can be
optically induced by interfering b e ams of o rdinary polar-
ization in experimental conditions. The probe beam is
extraordina rily polarized along the crystalline c axis. The
dimensionless variables x, y, and z are normalized to the
typical scales x
s
= y
s
= 1 µm, and z
s
= 1 mm, respec-
tively. The diffraction coefficient is D = z
s
λ/4πn
e
x
2
s
,
γ = 0.5k
0
n
3
e
γ
33
E
0
z
s
is a nonlinear coefficient propor-
tional to the electro-optic coefficient γ
33
and the applied
direct current (DC) field voltage E
0
, and K
0
is the wave
number. n
e
= 2.35 is the extraordinary refractive in-
dex of bulk photorefractive Strontium-Barium Niobate
(SBN) crystal, and λ = 532 nm is the laser wavelength
in vacuum. These parameters are selected so that they
are consis tent with a typical experimental setup
[9,24−27]
.
The dispersion relation and Bloch modes for high-
1671-7694/2013/090801(4) 090801-1
c
2013 Chinese Optics Letters