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Dipole and Vector Solitons in 2D Photonic Lattices. Jianke Yang Dept of Mathematics and Statistics, University of Vermont Igor Makasyuk, Anna Bezryadina, Zhigang Chen Dept of Phys. & Astronomy, San Francisco State University. Discrete solitons in waveguide arrays. - PowerPoint PPT Presentation
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Jianke YangJianke YangDept of Mathematics and Statistics, University of Vermont
Igor Makasyuk, Anna Bezryadina, Zhigang Chen Igor Makasyuk, Anna Bezryadina, Zhigang Chen Dept of Phys. & Astronomy, San Francisco State University
Dipole and Vector Solitons in 2D Photonic Lattices
Discrete solitons in waveguide arraysDiscrete solitons in waveguide arraysDiscrete solitons in waveguide arraysDiscrete solitons in waveguide arrays
D. N. Christodoulides et al., , Optics Letters 13, 794 (1988).H. S. Eisenberg et al., , Physical Review Letters, 81, 3383 (1998).
Optically-induced lattices Optically-induced lattices in photorefractive crystalsin photorefractive crystalsOptically-induced lattices Optically-induced lattices in photorefractive crystalsin photorefractive crystals
v
SBN
Efremidis et al., PRE 2002Fleischer, et al., PRL, Nature 2003Nashev, et al., OL 2003
From multiple o-beam interference Linear waveguides
Spatial modulation of a partially coherent o-beam
O-beamE-beam
Amplitude mask
Chen, et al. PRL2004
So far, fundamental and vortex solitons in a 2D lattice have been reported:
Fleischer, et al., PRL, Nature 2003Martin, et al., PRL 2004
Malomed and Kevrekidis, PRE 2001Yang and Musslimani, OL 2003Neshev, et al., PRL 2004Fleischer, et al., PRL 2004Yang, New J. Phys. 2004
In this talk, we report both theoretically and experimentally
dipole and vector solitons
in a 2D photonic lattice
Dipole solitons in a 2D latticeDipole solitons in a 2D lattice
Theoretical model:
,0||),(12
1)(
2
12
033
302
2
2
2
1
UUyxI
Ernk
y
U
x
U
kz
Ui e
D
y
D
xII
220 sinsin
Here U: electric field; z: propagation distance; E0 : applied DC field; D: lattice spacing;
I0: lattice intensity; r33: electro-optic coefficient;
k0= 20; k1= k0 ne;
Out-of phase dipole-solitonsOut-of phase dipole-solitons
High intensity Moderate intensity Low intensity Lattice
In-phase dipole solitonsIn-phase dipole solitons
High intensity Moderate intensity Low intensity Lattice
always unstable
Note:
the above dipole solitons arise due to a balance of
discrete diffraction
nonlinearity, and
lobe interactions
They can not exist without the lattice.
Simulations of a pair of Gaussian beamsSimulations of a pair of Gaussian beams
Out-of phase
In-phase
Input Output
Low NL High NL High NL No lattice
Quadrupole solitonsQuadrupole solitons
Out-of-phase In-phase
Can be stable Always unstable
Dipole solitons: experimental resultsDipole solitons: experimental results
In Phase
Out ofPhase
Input
Low NL High NL High NL No lattice
Output
Anisotropic effect: out-of-phase caseAnisotropic effect: out-of-phase case
Input
Low NL Low NL High NLNo lattice with lattice with lattice
Output
These dipole solitons are robust against anisotropic effects
Anisotropic effect: in-phase caseAnisotropic effect: in-phase case
Input Output
Low NL Intermediate NL High NL
These dipole solitons are sensitive to anisotropic effects
Vector solitons in a 2D latticeVector solitons in a 2D lattice
If we make the two beams of the dipole incoherent,
and launch into the same lattice site,
then we can study vector lattice solitons
2D vector lattice solitons: experiment2D vector lattice solitons: experiment
Input Output
Expt.results
Num.results
Low NL High NL High NL Coupled Decoupled
Mutually Incoherent
2D vector lattice solitons: theory2D vector lattice solitons: theory
,0||||),(12
1)(
2
122
033
302
2
2
2
1
UVUyxI
Ernk
y
U
x
U
kz
Ui e
,0||||),(12
1)(
2
122
033
302
2
2
2
1
VVUyxI
Ernk
y
V
x
V
kz
Vi e
,sin),(,cos),( zizi eyxVeyxU
Vector solitons can be derived from scalar ones by a polarization rotation:
(x, y) : scalar lattice soliton;
: polarization
Scalar 2D lattice solitons have been studied before:
Yang and Musslimani, Opt. Lett. 2003
Efremidis, et al. PRL 2004
Dipole-like vector solitons in a 2D latticeDipole-like vector solitons in a 2D lattice
If we make the two beams incoherent, and launch into different lattice sites,
then we can study dipole-like vector lattice solitons
Comb. input Low NL High NL 1st comp. 2nd comp.
Expt.results
Num.results
ConclusionsConclusions
1. We have demonstrated the formation of dipole, quadrupole, vector, and dipole-like vector solitons in a 2D photonic lattice for the first time.
2. These solitons arise due to a balance of discrete diffraction, nonlinearity, and lobe interactions.
3. These solitons are stable in certain parameter regimes.
A scalar lattice soliton
They are stable in a large parameter space
Dipole-like vector solitons in a 2D latticeDipole-like vector solitons in a 2D lattice
If we make the two beams incoherent, and launch into different lattice sites,
then we can study dipole-like vector lattice solitons
Comb. input Low NL High NL 1st comp. 2nd comp.
Expt.results
Num.results