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